Simplify each expression. Write answers without negative exponent...

Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (27/64)^-4/3

Accepted Answer

Hey everyone welcome back in this problem. We are asked to solve the given expression and our answer should be written without negative exponents. The expression we're given is 243 divided by 1000 and for all to the exponent negative 6/5. So just rewriting what we're given. 243 divided by for all to the exponent negative 6/5. Alright when we have something like this can be kind of hard to interpret um an exponent that's kind of a fraction or decimal we want to do is we want to try to break this up into two different exponents. Okay so let's recall that if we have a base A to the exponent of two things multiplied together. M. N. Hey we can write this as the A to the exponents M sorry. Okay all to the exponent N. Okay so this is our power rule for exponents. So in our case we have this negative six multiplied by an exponent of 1/5. So we can break this up and write this S 243 Divided by a 1024 To the exponent. 1 5th All to the experiment -6. All right now we could have done this either way because when we multiply we can switch the order of the multiplication. So we could have had the negative six inside the brackets in the 1/5. Outside. Okay the reason I did it this way Is because we're going to try to find 1/5 root of 243 in 2024. Okay if we do it the other way we're taking a fairly big exponents six. Okay, a fairly big numbers. So the numbers are gonna get really really really big before we take this fifth root and kind of make them smaller again. Um so sometimes it's simpler to take the route first. Okay. Deal with smaller numbers and then work with the bigger exponent. All right, So we have this exponent 1/5. Now when we have a fraction to an exponent, okay that exponent applies to the numerator and the denominator. So we can write this as 243 to the exponent 1/ divided by 1000 and 24 To the exponent 1/5. Okay. And this is all to the exponent -6. Alright now recall if we have a to the exponent one over N. This is equivalent to the end route of a. Okay, I was just talking about how this is the 5th root. Okay so we have one over exponent of 1/5. So that's equivalent to the fifth root. So if we take the fifth root of 243 we get to the exponent five is equal to 243. Similarly in the denominator. If we take the fifth root of 1000 and 24 we get four. Okay so four to the exponents five gives us 1000 and 24. So we get 3/4 All to the exponent -6. All right now we have a negative exponent. Let's recall how we deal with negative exponents. If we have a to the experiment negative end, you can write this as one over A. To the exponent N. We put the entire term in the denominator and the exponent becomes positive. So applying that to our problem We had 3/4. Oops not three or four. That's what we had three or four. We get one over this whole thing. Okay, the exponent is applied to this entire 3/4 term. So when it flips to the denominator, that entire base has to move. Okay, so we have 3/4 To the experiment six. Okay now it's a positive experiment. All right. Just like we distributed this exponent 1/5 above. Okay. We have a fraction all to an exponent. Okay, so that exponents gonna apply to the numerator and denominator. So this is gonna give us one divided by 3 to the exponents six divided by to the exponents six. Alright now we have this fraction. Okay, and we can deal with this like a normal fraction if we have 1/3 over four. Okay this is equivalent to Or to the exponents 6/3 to the exponent six. That's just using our regular rules for dealing with fractions. So we end up with four to the exponent six divided by three to the exponents six. Okay, now four to the exponent six is going to be 3 to the exponents six is 729. And so we get the solution to the expression we were given as 4096 divided by 729. If we go back up to our answer choices, we see that that is going to correspond with answer choice. D. Thanks everyone for watching. I hope this video helped see you in the next one.

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (27/64)^-4/3

Key Concepts

Negative Exponents

Rational Exponents

Simplifying Fractions

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