Free CHSPE Math Practice - CHPSE Study Guide

if the volume of a cube is 8 cubic centimeters what is the length of the cube let's start with what we know the volume is 8 cubic centimeters the formula for volume of a cube is side cubed and we know what the volume is so substitute 8 centimeters cubed for volume so that equals the side cubed or the length cubed since it's a cube all of the dimensions are the same so if they asked for length we can find that it's the same as the width and the height which is why we just use an S instead of length times width times height but how do we solve so they have side cubed if we want to solve for side then we need to do the opposite of cubing which is to cube root so take the cube root of both sides when you're taking the cube root really what you're doing is you're finding what number times itself three times would give you 8 and that number would be 2 2 times 2 times 2 is 8 or 2 cubed is 8 so the cube root of 8 is 2 and the cube root of centimeters cubed is simply centimeters and that equals the cube root of side cubed which is side 4 again length or width or height all the dimensions are the same so what is the length answer be 2 centimeters simplify the following expression 16x cubed minus 32 x squared plus 8x all divided by 4x we need to divide each one of these terms by 4x so the first thing I'm going to do is rewrite this as three separate fractions and now we can simplify 16 divided by 4 is 4x cubed divided by X is x squared because when you divide you subtract the exponents 3 minus 1 is 2 minus 32 divided by 4 is 8 x squared divided by X 2 minus 1 is X to the first plus 8 divided by 4 is 2 and X divided by X or X to the first divided by X to the first 1 minus 1 is 0 or you can just think X or anything really divided by itself is 1 so X divided by itself is 1 and 1 times 2 is 2 so our answer is C solve for x in the following inequality 4x plus 23 is greater than negative 3x minus 6 the first thing we want to do is get all of our X terms on the same side of the inequality I'm going to do this by adding 3x to both sides 4x plus 3x is 7x bring down the plus 23 and bring down the greater than sign negative 3x plus 3x those are additive inverses so they cancel bring down the negative 6 next we want to remove 23 the opposite of adding 23 is subtracting 23 so subtract 23 from both sides bring down the 7 X plus 23 and negative 23 are additive inverses so they cancel bring down the greater than sign negative 6 and negative 23 they have the same signs they're both negative which means our answer is negative and we just add the numbers together 23 plus 6 is 29 so it's negative 29 finally we have to divide both sides by 7 7 divided by 7 is 1 and 1 times X is simply X bring down the greater than sign and then the first thing I'll do is change this into a mixed number since I can't actually divide negative 29 by 7 and get an integer so 7 goes into negative 29 4 times so it's negative 4 4 times 7 is 28 so that means that there's one leftover out of 7 so it's negative 4 and 1/7 X is greater than negative 4 and 1/7 now if we look at the answers you'll see that all the answers are in decimal form and we can find out what one 7th is as a decimal by dividing so seven doesn't go into one so we'll have to put a decimal here which means bring a decimal to the top and add a zero to it seven does go into ten one time seven times one is seven we subtract and we get three add another zero bring it down and seven goes into 34 times 7 times 4 is 28 we subtract and we get two and we can stop there because all of our answers stop at the hundredths place as well so we could write this as X is greater than negative 4 and 14 hundredths which is answer choice a what is the value of six factorial what six factorial means is 6 times 5 times 4 times 3 times 2 times 1 and since we're just multiplying a series of numbers you can multiply them in any order you want so you can take any two numbers and multiply those together first and then continue multiplying but I'll just multiply them in order so we have 6 times 5 which is 30 and times 4 times 3 times 2 times 1 then we can multiply 30 times 4 we have 30 60 90 120 so 30 times 4 is 120 times 3 times 2 times 1 then 120 times 3 120 times 3 3 times 0 is 0 3 times 2 is 6 3 times 1 is 3 so 360 times 2 times 1 360 times 2 2 times 0 is 0 2 times 6 is 12 carry the 1 2 times 3 is 6 plus 1 is 7 so 720 times 1 and 720 times 1 is just 720 so 6 factorial is equal to 720 answer see simplify the following expression the quantity 2x minus 20 times the quantity 5x plus 10 to multiply these two binomials we can use a method called foil and foil stands for first o stands for outer I stands for inner and L stands for last so the F first means to multiply the first term in the first binomial times the first term in the second binomial first times first 2 x times 5 X 2 times 5 is 10 and x times X is x squared so it's 10 x squared the again stands for outer so that means to multiply the two outside terms so 2x and 10 are the two terms on the outside the outer terms so 2x times 10 2 times 10 is 20 2 x times 10 is 20 x I stands for inner that means to multiply the two terms on the inside so that would be this negative 20 times 5x negative 20 times 5x a negative times a positive is a negative 20 times 5x is 100x then finally we multiply the last term's together so that means the last term in the first set of parentheses or in the first binomial times the last term in the second binomial last times last so that's negative 20 times 10 which is negative 200 negative times a positive is a negative we're almost done the last thing we need to do is combine these like terms right here 20x and negative 100 X so it's really 20x minus 100x bring down the 10x squared first and 20x minus 100x would be negative 80 X then bring down the negative 200 so looking at our answer choices looks like answer a would be the correct answer solve for X using the following system of equations 3y plus 10x equals 23 18 y minus 15 x equals 50 there are many different ways to solve a system of equations one is by graphing you would graph the two lines and then see where they intersect the solution is the ordered pair of the intersection of those lines because the intersection is the only ordered pair that will satisfy both equations and make them both true since it's the only point they have in common you can also use substitution that's where you solve for one variable and then substitute that expression into the other equation and then there's elimination and that's the method we're going to use now to solve for X since we want to solve for X we want to eliminate Y and eliminate means to get rid of to get rid of Y I need to multiply this whole equation by a negative 6 because negative 6 times 3y is negative 18 y and then when I add my two equations together 18 y and negative 18 y will cancel or eliminate negative 6 times 10 x is negative 60 X and negative 6 times 23 is negative 6 times 20 is 120 6 times 3 is 18 so 1 38 and now I need to add my two equations together 18 Y and negative 18 1 is 0 they cancel negative 15 X and negative 60 X is negative 75 x equals negative 1 38 and 50 we're going to add those together I'm going to get a negative and I can subtract off to the side here that's 8 and 13 minus 5 is 8 so negative 88 finally to solve for X we need to divide both sides by negative 75 a negative divided by a negative is a positive and 75 divided by 75 is 1 times X is X and again a negative divided by a negative will give me a positive we can divide those off to the side 75 goes into 88 one time 75 times one is 75 we subtract and we get 13 add a decimal and a zero bring it down 75 goes into 130 one time 75 times one is 75 subtract and we get 55 add a zero and bring it down and seventy five goes into 550 I believe seven times seven times seventy is 490 seven times five is thirty-five said be 525 we subtract and we get 25 add a zero bring it down and we could keep on going dividing until we find the end but since this is multiple choice we can already see which answer choice is correct and that would be answer B 1 + 17 hundredths if 2x plus 5x equals 3x plus X plus 30 what is the value of X so first I'm going to combine like terms on both sides of the equal sign on the left side I can combine 2x plus 5x to give me 7x bring down this equal sign and on the right side we can combine 3x plus X so 3x plus another X gives us 4x and then bring down the plus 30 next we need to get all of our X terms on the same side of the equal sign we'll do this by subtracting 4x from both sides 7x minus 4x is 3x bring down the equals 4x minus 4x cancels they're additive inverses and we bring down the positive 30 finally we need to divide both sides by 3 3 divided by 3 is 1 and 1 times X is simply X so x equals 30 divided by 3 which is 10 and that's answer choice D nine X minus three y plus eight X Y minus three if x equals ten and y equals negative two what is the value of this expression first we need to substitute nine times X and we're replacing X with 10 so 9 times 10 minus 3 times y and we're replacing Y with negative 2 plus 8 times X times y so times 10 times negative 2 minus 3 now we need to follow our order of operations first would be grouping symbols but in this case there aren't any next would be exponents and again we don't have any so after that is multiplication and division working from left to right starting at the left we have 9 times 10 which is 90 minus 3 times negative 2 plus 8 times 10 times negative 2 minus 3 our next set of multiplication would be right here so I'm going to bring down my 90 minus 3 times negative 2 is negative 6 plus 8 times 10 times negative 2 minus 3 our next set of multiplication would be 8 times 10 so we have 90 minus a negative 6 plus 8 times 10 is 80 times negative 2 minus 3 then we need to multiply 80 times negative 2 so we bring down our ninety minus negative six plus eighty times negative to a positive times a negative is a negative so negative 160 minus three now that we've taken care of all the multiplication we're now going to work with addition and subtraction again from left to right so starting on the Left we're going to start with ninety minus a negative six minus a negative you can think of this like you're taking away a negative which is the same as adding a positive or you can add the inverse so we have 90 plus six 96 plus negative 160 minus three again we've got to continue working from left to right so next we have 96 plus negative 160 these two numbers have different signs 96 is positive while 160 is negative since they have different signs we need to subtract them so I'll do that off to the side 160 minus 96 so to borrow from the six it becomes a 5 10 minus 6 is 4 now borrow from the 1 it becomes a 0 so I have 15 minus 9 is 6 so 160 minus 96 is 64 and it's going to be a negative 64 because 160 or negative 160 has a larger absolute value than 96 tose so negative 64 minus 3 and the last thing we do is subtract those numbers so if you think about where you are in the number line you're at way over here on the left at negative 64 and then you subtract 3 so you're moving more to the left your numbers getting even smaller so it's negative 67 another way to think about it is you have 64 negatives and three negatives that's a total of 67 negatives and finally even another way to think about it is you can change it to adding the inverse so you have negative 64 plus negative 3 either way you're going to get negative 67 and that's answer a if a rectangle has a length of 5 centimeters and a width of 7 centimeters what is its area we're finding the area of a rectangle the formula for area of a rectangle is area equals the length times the width and we were given the length five centimeters and we were given the width seven centimeters so all we have to do is substitute and solve the area is length is five centimeters times the width which is seven centimeters so the area is five times 7 is 35 centimeters times centimeters is centimeters squared so the area is 35 centimeters squared which is answer choice B in the following inequality solve for X negative 4x plus 8 is greater than or equal to 48 to solve for X we need to get X alone which means undoing the operations that are being done to X currently and that is negative 4 times X and plus 8 so we'll start by getting rid of the +8 and we can get rid of it by doing the exact opposite of adding 8 subtracting 8 and if we subtract 8 from the left side of our inequality symbol then we must also subtract 8 from the right side to keep it even so we bring down our negative 4x a positive 8 minus 8 is 0 is greater than or equal to 40 8 minus 8 is 40 we're almost done now we need to undo negative 4 times X and the opposite of multiplying by a negative 4 is dividing by a negative 4 now with inequalities there is a special rule you have to remember and that is if you divide both sides by a negative number or you multiply both sides by a negative number you have to flip the sign so now instead of greater than or equal to our sign is now a less than or equal to sign and again that's because we divided both sides by a negative number negative 4 divided by negative 4 is 1 and 1 times X is X ford e / negative for a positive divided by a negative is a negative and 40 divided by 4 is 10 so X is less than or equal to negative 10 answer d a classroom contains 13 boys and 18 girls if a student's name is chosen randomly what is the probability it will be a girl's name to find probability we need to write a fraction that fraction will be how many girls names there are out of how many total names there are so since there are 18 girls that means we have 18 girl names to choose from out of now we have to add in the 13 boys to get our total so 18 plus 13 is 31 so 18 names out of a total of 31 names are going to be girls if we look at our answers though our answers are not in fraction form our answers are all in percentage so we need to convert our fraction into a decimal and then into a percent so to convert our fraction into a decimal we just divide 18 divided by 31 31 can't be divided into 18 but we can add a decimal and a 0 31 does go into 185 times 31 times 5 is 155 borrow from the 8 becomes a 7 then we have 10 minus 5 which is 5 and 7 minus 5 which is 2 then we add a 0 and bring it down and 31 goes into 258 times 31 times 8 is 248 we subtract and we get 2 then you can add a 0 and bring it down but 31 doesn't go into 20 so we get 0 goes 20 0 times now we can stop here because if you look at your answers they don't go beyond two numbers and to change our decimal into a percent we just move our decimal two places to the right so fifty eight hundredths is equal to 58 percent which is answer C the quantity y plus 10 squared minus 625 equals 0 solve for y the first thing we want to do is add 625 to both sides because doing that is going to help us get Y alone and thus solve for it so we bring down our quantity y plus 10 squared negative 625 plus 625 is 0 so that just cancelled out and that equals 0 plus 625 just 625 well then we want to get rid of this squared because our Y is kind of locked in those parentheses so to unlock them we want to do the opposite of squaring which is square rooting so we square root both sides and the square root and the square undo each other they're the exact opposites of each other so they cancel each other out and we're just left with y plus 10 equals and when I take the square root of 625 I get plus or minus 25 and that's because a positive 25 squared would give me 625 but also a negative 25 squared would give me 625 so we really have two answers there so we write it with a plus or minus and finally to solve for y we need to subtract 10 from both sides so we have y plus 10 minus 10 is 0 so equals negative 10 plus or minus 25 we have two answers and to find them we just need to simplify so we have y equals negative of ten plus 25 and y equals negative ten minus 25 all I did was separate the plus or the minus into two separate expressions negative 10 plus 25 is 15 while negative 10 minus 25 is negative 35 so our solutions are that Y is 15 or negative 35 which is answer D if y equals to what is the value of the following expression the quantity Y to the ninth divided by Y cubed times 2 the first thing we need to do is substitute 2 for Y in our expression so it becomes 2 to the ninth divided by 2 cubed and that quantity times 2 and what I want to do is rewrite it I prefer my fractions to be stacked so I'm going to rewrite this as 2 to the ninth divided by 2 cubed times 2 when we're dividing and we have our bases the same so in this case 2 is our base and it's the same we simply subtract our exponents 9 minus 3 is 6 so that's 2 to the 6 times 2 2 to the 6 is 2 times 2 times 2 times 2 times 2 times 2 so 2 times itself 6 times and then times 2 so all we have to do is multiply all these twos 2 times 2 is 4 2 times 2 is 4 2 times 2 is 4 and then times 2 4 times 4 is 16 times and 4 times 2 is 8 so now I just need to multiply 16 times 8 8 times 6 is 48 carry the 4 8 times 1 is 8 plus 4 is 12 so it's 128 factor the following expression 9x squared Y minus 18 XY minus 27y the first thing we need to do when factoring is to look and see if there's a GCF for our factor for our terms so I'm going to write down our terms 9x squared Y 18 XY and 27 Y and then I'm going to factor them to see if I can find a GCF the prime factorization of 9 is 3 times 3 and then x squared so that's times X times X and then times y so that is the prime factorization of this term 18 is 2 times 9 and 9 is 3 times 3 times X times y 27 is 3 times 9 and 9 is 3 times 3 times y now to find the GCF we look for what all three of our terms have in common all three of our terms have a 3 in common all three of our terms have another 3 in common and all three of our terms has a Y in common so the GCF is 3 times 3 times y 9y now we can factor out the GCF so we're going to factor out a 9 Y from each one of our terms 9y times x squared would be 9x squared y 9y times negative 2x would be negative a eighteen XY and nine y times negative 3 would be negative 27 y now we need to look and see if we can factor this trinomial so bring down the 9y and if this trinomial can be factored we're going to factor it into two binomials well I know that I'm going to have to have X and X as my first term in each set of parentheses because x times X is x squared now the numbers that go here have to multiply to give me negative 3 and add to give me negative 2 so we need numbers that we the factors of negative 3 that have a sum of negative 2 put that in parentheses the factors of negative 3 that have a sum of negative 2 and the only factors of negative 3 that would have a sum of negative 2 would be negative 3 and positive 1 so negative 3 and positive 1 negative 3 times 1 is negative 3 and negative 3 plus 1 is negative 2 and we can always foil it out to see if it does in fact we did factor correctly so x times X would give us our x squared x times 1 is 1 X - 3 X would be negative 2 X and then negative 3 times 1 would be negative 3 so this is our factored answer which is answer choice C if W equals seven calculate the value of the following expression 8w squared minus 12 W plus the quantity of 4w minus five plus six in order to simplify the first thing we need to do is substitute seven for W so we start by bringing down the eight we're going to replace W is seven so it's 8 times 7 squared minus 12 again replace W plus seven so times 7 plus bring down the parentheses 4 times W or times 7 minus 5 plus 6 in order to simplify this we must follow the order of operations or PEMDAS we start with parentheses then exponents then we multiply and divide from left to right and finally we add and subtract from left to right so that means we need to start with anything inside parentheses so here we have a couple of different operations inside our parentheses we have multiplication and subtraction well again if we consult PEMDAS we can see that we are supposed to always multiply and divide before we add and subtract so that means we're starting with four times seven so you'll just copy everything else the eight times seven squared minus twelve times 7 plus bring down the parentheses finally we get to 4 times 7 28 minus 5 and bring down the plus 6 so we haven't finished simplifying what's in our our parentheses and that's what we'll do now 28 minus 5 so bring down the 8 times 7 squared minus 12 times 7 plus 28 minus 5 is 23 plus 6 now if we go back to PEMDAS after parentheses which we have no more of we then simplify any exponents and we do have exponents right here 7 squared so that's what we'll do next 8 times 7 squared means 7 times 7 and that's 49 minus 12 times 7 plus 23 plus 6 okay after exponents is our multiplication and division from left to right so first we see 8 times 49 we can just go off to the side and multiply that 49 times 8 8 times 9 is 72 so write the 2 and carry the seven eight times four is thirty-two thirty-two plus seven is 392 so we have three hundred ninety-two minus 12 times seven plus 23 plus six we need to continue multiplying and dividing from left to right so next I see 12 times seven so we'll multiply that 12 times 7 7 times 2 is 14 all right the 4 carry the 1 7 times 1 is 7 plus 1 is 8 that's 84 bring down the 392 minus 84 plus 23 plus 6 so all we have left now is our addition and subtraction from left to right so they start with 392 minus 84 I'll do that over here so we have to borrow from the 9 it becomes an 8 we have 12 here 12 minus 4 is 8 8 minus 8 is 0 and we bring down the 3 so we have 308 plus 23 plus 6 so that means we need to add 23 to our 308 that's the next thing we're going to do 8 plus 3 is 11 carry the 1 1 plus 2 is 3 and bring your 3 down so we have 331 plus 6 that's the last thing we're going to do is add 6 to 331 so we get 7 3 3 our final answer is 337 which is answer choice C you simplify the following expression the square root of 3 times the quantity 5 square roots of 3 minus the square root of 12 plus the square root of 10 since there is no operator between the square root of 3 and the parenthesis like there's no addition sign or subtraction sign or division sign it's understood to be multiplication so I'm multiplying the square root of 3 times all of these terms inside my parentheses which means we're going to use the distributive property so I'll start by multiplying the square root of 3 times 5 square roots of 3 so that's 5 times and then the square root of 3 times the square root of 3 is the square root of 9 which is 3 so that's 5 times 3 the square root of 3 times the negative square root of 12 is the negative square root of and you just multiply your numbers under the square roots of 3 times 12 is 36 and then the square root of 3 times the square root of 10 so again we're just multiplying our numbers under the radical 3 times 10 is 30 so the square root of 30 but we're not finished yet because we have some simplifying that we can do 5 times 3 is 15 minus the square root of 36 is 6 because 6 times 6 is 36 plus now 30 you want to break down that number and see if there is any part of it that we can take the square root of so to break it down you can just do a factor tree 30 is 3 times 10 3 is prime and 10 is 2 times 5 and those are both prime so I have no pairs of numbers and no perfect square factors so I can't take the square root of any part of 30 so it has to be left is just the square root of 30 and finally we can subtract 15 minus 6 since those are like terms and we get 9 so 9 plus the square root of 30 so our answer is a simplify the following expression 2x to the fourth Y to the 7m squared Z times 5x squared Y cubed M to the 8th I'm going to start by multiplying my like terms together so I can rewrite this it's multiplication so you can multiply in any order you want so I'm going to multiply two times five together and then I've got this X to the fourth times the x squared and then we have y to the seventh times y cubed and then we have M squared times M to the eighth and then finally we just have this Z so times Z and now we'll simplify each one of these so 2 times 5 is 10 and so right now you could already eliminate answer choice B because we're multiplying we're not adding two and five together we're multiplying and then we get to our X's X to the fourth times x squared what that really is X to the fourth means x times itself four times while x squared means x times itself two times so when we multiply X to the fourth times x squared we're multiplying x times itself a total of six times so it's X to the sixth you can see what we end up doing is adding those exponents together and it's the same thing with Y to the seventh times y cubed let's look at that one a little differently so Y to the seventh times y cubed why did the seven is why times itself 7 times 1 2 3 4 5 6 7 times y cubed so times 1 2 3 so what we end up with is y times itself a total of 10 times so Y to the seventh times y cubed is y to the tenth or again we're adding those exponents then we have M squared times M to the eighth so same concept again we're going to add our exponents we get M to the tenth and then finally times Z so that means our answer is d 10 X to the sixth Y to the 10th into the tenth Z solve for y using the following system of equations 2x minus 6y equals 12 negative 6x plus 4y equals 42 there are several ways to solve a system of equations what you need to keep in mind is that what you're doing is finding where these two lines intersect on a graph so you can graph that's one way to solve a system is to graph both the lines and see where they intersect and then in this case just give the y coordinate but you can also use substitution or elimination and it's really your choice which one seems the best for the situation in this situation elimination seems to be the easiest choice because if I multiply this top equation by 3 I'll have a positive 6 X term and a negative 6 X term that will eliminate thus allowing me to solve for y so I'm going to multiply this entire top equation by 3 3 times 2x is 6x 3 times negative 6 y is negative 18 1 and 3 times 12 is 36 now I'm just going to add these two equations together negative 6x and 6x is 0 they cancel 14 Y and negative 18 Y is negative 4y bring down the equals 42 plus 36 is 78 then to solve for y I need to divide both sides by negative 4 negative 4 divided by negative 4 is 1 times y is y equals a positive divided by a negative is a negative and then I just need to go off to the side and divide 78 by 4 4 goes into 7 one time 4 times 1 is 4 he's subtracting it 3 spring down the ace 4 goes into 38 9 times 4 times 9 is 36 subtract and you get to add a decimal add a decimal and a 0 bring it down and 4 goes into 20 5 times 4 times 5 is 20 subtract and you get a remainder of 0 so it's negative 19 and 5 tenths which is answer a the length of a square is 15 centimeters what is its area well a square is a type of rectangle and the formula for area of a rectangle is area equals the length times the width or base times height but what's special about a square is that its length and width are the same so if we're told that the length is 15 centimeters then that means the width is 15 centimeters as well so we substitute 15 centimeters for the length and 15 centimeters for the width so 15 times 15 15 times 15 can multiply that out over here five times five is 25 right the 5 carry the two five times one is five plus two is seven get rid of that put your zero placeholder one times five is five and one times one is one then we add these up five and zero is five 7 plus 5 is 12 right the two carry the 1 + 1 + 1 is 2 so 15 times 15 is 225 and centimeters times centimeters is centimeters squared so the area of a square with length 15 centimeters is 225 centimeters squared which is answer choice D if X divided by three plus seven equals 35 what is the value of X I'm going to start by writing this problem X divided by 3 plus 7 equals 35 the first thing we want to do is subtract 7 from both sides because a positive 7 and a negative 7 are additive inverses of each other and they cancel so on the left side were left with X divided by 3 and on the right side 35 minus 7 is 28 the opposite of dividing by 3 is multiplying by 3 so now we need to multiply both sides by 3 if you put this 3 over 1 now you're multiplying two fractions together and the numerator and this denominator cross cancel and you're simply left with X divided by 1 which is just X so now if we determine what 28 times 3 is we'll have our answer 3 times 8 is 24 so write the 4 and carry the 2 and 3 times 2 is 6 plus 2 is 8 so that means that X is 84 which is answer C if 2x equals 5x minus 30 what is the value of x the first thing we want to do to solve for X is to get all of our X terms on the same side of the equal sign we have two X on the left side of the equal sign and we have 5x on the right side of the equal sign we have two options here we can either subtract 2x from both sides and that way we would get rid of 2x on the left side or we can subtract 5x from both sides and get rid of 5x on the right side I'm going to subtract 5x from both sides this is called the subtraction property of equality because I have the left side equal to the right side then I can subtract the same thing from both sides and they'll still be equal to X minus five X is negative three X we bring down our equal sign 5x minus 5x is zero because those are additive inverses so we've cancelled that out and what we have left on the right side is negative 30 now we have negative 3 times X is equal to negative 30 so to solve for X when you do the opposite of multiplying by a negative 3 which is dividing both sides by a negative 3 negative 3 divided by negative 3 is 1 and 1 times X is just X so we have x equals negative 30 divided by negative 3 a negative divided by a negative is a positive and 30 divided by 3 is 10 so x equals 10 which is answer choice a a rectangular prism measures 12 centimeters by 3 centimeters by 9 centimeters what is its volume the formula for volume for a rectangular prism is volume equals length times width times height and we were given those dimensions length width and height and really it doesn't matter which one is the length which one's the width and which one's the height since we're just going to multiply those three numbers together anyway so the volume is 12 centimeters times 3 centimeters times 9 centimeters and it's up to you which to you choose to multiply together first good start with 12 times 3 3 times 2 is 6 and 3 times 1 is 3 so 12 centimeters times 3 centimeters is 36 centimeters squared times 9 centimeters so then you could take 36 times 9 9 times 6 is 54 9 times 3 is 27 plus 5 is 32 so 36 centimeters squared times 9 centimeters is 324 centimeters cubed which is answer choice C on a six-sided die each side has a number between one and six what is the probability of throwing a three or a four this die has six sides that means our total is six which is the denominator of our fraction for finding our probability we want to know what the probability of throwing a three or and that is a keyword three or a four so if it lands on either one of those were good so that's two different numbers that we could land on out of our six different numbers and then of course we can simplify our fraction by dividing the numerator and the denominator both by 2 2 divided by 2 is 1 and 6 divided by 2 is 3 so it's 1 in 3 or written like this a 1 in 3 chance simplify the following expression the quantity 2x squared plus 3 times the quantity 2x minus 1 to simplify we need to foil the F in foil stands for first so we're going to take the first term in our first binomial and multiply it times the first term in our second binomial first and first 2 x squared times 2 X 2 times 2 is 4x squared times X is X cubed next we do au which stands for outer so those are the two terms on the outside which would be 2x squared and negative 1 so 2x squared times a negative 1 is negative 2x squared then our inner terms or the two terms on the inside 3 times 2x which is a positive 6x and then finally our last terms so the last term in the first set of parentheses and the last term in the second set of parentheses 3 times negative 1 which is negative 3 then we want to look and see if we had any like terms to combine and we don't we have a cubic X term a squared X term and X term and then just a constant so that's our answer which matches answer choice a simplify 9x times the quantity 3x squared plus 2x minus 9 to simplify we need to distribute the 9x to all the terms inside the parentheses and what that means is we have to multiply so we multiply first 9x times 3x squared 9 times 3 is 27 x times x squared is X cubed plus 9x times 2x 9 times 2 is 18 x times X is x squared minus 9x times 9 9 times 9 is 81 X and that would be answer choice B 27 X cubed plus 18x squared minus 81 X so remember that when you multiply x times x squared you're adding the exponents and that's where the X cubed comes from x times X again you're adding those exponents and those have an imaginary X 1 or double 1 both of them so X to the first times X to the first is x squared you add the exponents 1 plus 1 and the 9x times 9 is 81 X or negative 81 X because of the minus in front of the 9

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Length: 65min 25sec (3925 seconds)

Published: Tue May 03 2016