this article is for those of you who are going to take algebra so this is just an introductory course let's go over the basics let's talk about multiplication if you want to multiply 5 times 4 what's the answer now perhaps you know the answer by memorization when you think of multiplication think of repeated addition 5 times 4 means that you're adding 5 4 times five plus five is ten and these two fives add up to ten ten plus ten is twenty so that's multiplication it's just repeat addition so if you wanna multiply eight times three basically you're adding eight three times eight plus eight is 16 sixteen plus eight is twenty four division is the opposite of multiplication for example we said that 8 times 3 is 24 therefore 24 divided by 3 is 8. now let's say if you want to divide 48 by 4 what is that equal to so ask yourself four times what number is forty eight if you know the multiplication tables four times 12 is forty eight you can also use long division to get the answer as well whenever you need to divide fractions four goes into four one time bring down the eight four goes into eight two times and so the remainder is zero so 48 divided by four is 12. now let's review exponents what is four raised to the third power what does that mean when you're dealing with exponents you're dealing with repeated multiplication four to the third power means that you're multiplying three fours together four times four times four is sixty-four so five squared means that you're multiplying five two times five times five is twenty-five now what is the difference between negative three squared and negative three squared so in a typical algebra class make sure you understand the difference between these two negative three squared in parenthesis means that you're multiplying two negative three values a negative times a negative is a positive number three times three is nine now negative 3 squared without the parentheses means that you have one negative sign and you're multiplying two positive three values so overall this answer is negative nine in addition anything raised to the zero power is one now when dealing with variables when you multiply common variables you can add the exponents x squared times x cubed is x to the fifth so basically x squared means that you're multiplying two x values x cubed three x values so in total you have five x variables being multiplied together now what is x squared raised to the third power when you raise one exponent to another exponent you need to multiply two times three is six so this expression means that you have x squared times x squared three times which each x squared is x times x so in total you're multiplying six x variables and that's why it equals x to the sixth now what about dividing let's say x to the seventh divided by x cubed what is that equal to so when you're dividing what you need to do is subtract the exponents seven minus three is four to understand it x to the seven equates to seven x variables being multiplied together x cubed represents three x variables so if you cancel three on top three on the bottom notice that you'll have four left over on top so x to the fourth now you need to be familiar with negative exponents x to the negative three is the same as one over x cubed so if you move the x variable from the top to the bottom the exponent changes sign it changes from negative three to positive three so x to the negative four is one over x to the fourth likewise if you have a negative exponent on a bottom you can move the x variable to the top and this will become positive x squared so let's say if you want to multiply 4x cubed by x to the sixth let's multiply four times five four times five is twenty and then we need to multiply x cubed times x to the sixth which we need to add three and six that's nine so it's 20x to the ninth so go ahead and work on these examples five x to the fourth times three x to the seventh and 24 x to the eight divided by six x to the fifth so five times three is 15 and x to the fourth times x to the seventh we need to add four plus 7 so that's 11. here we need to divide 24 divided by 6 is 4. and x to the 8 divided by x to the 5th we got to subtract 8 minus 5. so it's 3 so you get 4 x cubed here's another example simplify this expression 28 x to the fourth divided by 35 x to the seventh feel free to pause the video now in this example i'm going to break down 28 into 7 and 4. 35 i'm gonna break it down into seven and five x to the seventh I want to write it as x to the fourth times x to the third because four plus three adds up to seven notice that I can cancel 7 and I can cancel x to the fourth so the final answer is 4 divided by 5 x cubed now granted you could have subtract four and seven four minus seven is negative three now it's x to the negative three on top but if you move it to the bottom it becomes one over x cubed so in the end you should have x to the third on the bottom here's another example that's similar let's say if we hav e 63 x to the fourth y to the third over 56 x to the eighth y to the fifth 63 is nine times seven and fifty-six is eight times seven x to the eighth is x to the fourth times x to the fourth y to the fifth is y squared times y cubed two plus three is five four plus four is eight and then i'm gonna cancel the seven i'm gonna cancel x to the fourth and i'm also gonna cancel y cubed so my final answer is 9 divided by 8 x to the fourth y squared so that's a nice simple way to simplify uh expressions that look like that now sometimes you may have to deal with exponents let's say we have two x cubed y to the fourth raised to the third power how can we simplify this expression so when you raise one exponent to another you need to multiply so first we're going to multiply three by one three times one is three and then three times three which is nine and four times three which is 12 two to the third power means that we're multiplying three twos together two times two times two is eight so the final answer is eight x to the ninth y to the twelfth try this one three x squared y to the third raised to the second power times four x cubed y to the fourth raised to the third power so let's distribute two two times one is two so we're just going to have three squared and then two times two is four so that's gonna be x to the fourth and three times two is six and then we have three times one which is three and then three times three that's nine and finally three times four is 12 three squared is three times three four cubed is four times four times four x to the fourth times x to the knife we need to add four plus nine so that's 13 and y to the sixth times y to the 12 we have to add 6 and 12 which is 18. now 3 times 3 is 9 4 times 4 times 4 is 64. so what is 9 times 64 if you don't want to use the calculator here's what you can do you can multiply sixty four by ten minus one because ten minus one is nine sixty four times ten is six forty and sixty four times one is sixty four now i'm going to write 64 as 40 and 24 because 40 plus 24 is 64. so negative 40 negative 24 is negative 64. 640 minus 4600 and if you take away 24 from 600 you can view 24 as 20 and 4. 600 minus 20 is 580 and 580 minus 4 is 576. if you want to subtract things in your head that's how you can do it so this is 576 x to the 13 y to the 18th
Introductory Algebra For College Students
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