Match each expression in Column I with its equivalent expression ...

Question

Match each expression in Column I with its equivalent expression in Column II. See Example 8.  a. (4/9)^3/2 b. (4/9)^-3/2 c. -(9/4)^3/2 d. -(9/4)^-3/2  A. 27/8 B. -27/8 C. 8/27 D. -8/27

Accepted Answer

Hey everyone in this problem we are asked to choose the option from column B. Which is a simplified form of the expression in column. Okay so we have column A. Here on the left. Okay we have four different expressions A, B, C. And D. We want to match those to these simplified expressions in column B. On the right. Alright so let's start with A. Okay, we're gonna start with column A. Capital A. And then we're just gonna work one at a time. So starting with a little A. Okay, expression were given is 32 divided by 243 to the exponent for over five. No, when we have an exponent like this we can think of this as two things multiplied together. We can think of this as an exponent four times an exponent of 1/5. 2 things multiplied together. And recall when we have two things multiplied together like this. Okay, we are powerful for exponents and we can write this as 32 divided by 243 to the exponents. 1/5 and then all of this to the exponent four. Okay, so we can split those two exponents and do them one at a time. Now you can do this in either order. Okay we're talking about multiplication of those exponents. So we know that multiplication, we can switch the order. So we can swap the order of these and have the four inside the brackets and the exponent of 1/ outside of the brackets. Okay we've chosen to do it like this. Um Because if we take an exponent four of 32 243 those numbers are gonna start to get really, really big before we take this exponent of 1/5. If we do it the opposite way the way that we've chosen to do it, the numbers are gonna get smaller first before we have to do that exponent of four, so it's just a little bit easier to work with. Alright now using our rules, okay, we know that if we have a fraction to an exponent, okay that that exponent applies to both the numerator and the denominator. So we can write this as 32 to the exponent 1/ divided by 243 to the exponent 1/5. And all of this is still to the exponent for. So now instead of doing an entire fraction to an exponent we just have to do each part. Now when we have an exponent of 1/5 this is like taking the fifth root. Okay this is equivalent to taking the fifth root, so what's the fifth root of 32? Well the fifth root of 32 is two. Okay because two to the exponents five is equal to and the fifth root of 243 is three. Okay because three to the exponent five is 243. When the brackets we end up with two thirds and this is all to the exponent four. Now just like we did when we distributed the exponent of 1/5. We can do the same with the exponent of four. Okay so this is going to be equal to the exponent of two to the exponent four. Or sorry to to the exponent four divided by three to the exponent four. We know that two to the exponent four is equal to 16 and three to the exponent four is equal to 81. And so for part A we get that are simplified expression is going to be 16/81. Mhm. So let me erase this here and I'm just gonna draw lines connecting these answers. Okay? So we found that for a this is gonna match with answer C. All right. Let's move to be. Okay. So for B what do we have? Well we have 32 to the X. Or sorry, divided by 243 to the exponent negative forfeit. Hm Alright so we're gonna do the same thing that we did before. We're gonna break this up into an exponent of negative four, multiplied by an exponent of one. Fit Our power rule allows us to separate those and deal with one exponents at a time. So we can write this as 32 divided by 243 to the exponent 1/5 all to the exponent negative four. Now we've done 32 divided by 243 to the exponent 1/ above. Can we know that? That's equal to two thirds. So we're not going to write out all the steps again. We're gonna skip that part because we've just done it and we know that this is equal to two thirds and then we have our exponent negative four. Alright, now when we have a negative exponents were called that we can flip this. Okay, we can put the entire base and exponent into the denominator and the exponent will become positive. Okay? So if we do that we're gonna get one divided by to over three to the exponent positive four when we have a fraction applied to an exponent nursery an exponent applied to a fraction, we can apply it to the numerator and denominator separately, so we can write this as 1/2 to the exponent four divided by three to the exponent four. Now, from our regular rules for dealing with fractions, okay, we know that this is going to be equal to 3/4 divided by 2/4. We know that 3/4 is going to be 81 2/4 is going to be 16. Can we use those values when we looked at part A. And so we get that for part B. The simplified expression is going to work out to 81/16. Alright. And what you'll notice is we had the exact same magnitude of our exponents in part A and part B. A. We had four faces and B. We had negative for fists. Okay and what we get is just kind of the inverse of the answer. Okay? We had 16/81. Now we get 81/16. Okay so you can see that relationship between the positive and negative exponent. Alright so for part B we found that we are equal to 81/16 and if we match this we are going to match this with answer. Alright so we're halfway there. Now let's move on to C. And D. Move down and work down here. Alright so for part C the expression were given is negative. Okay we're negative is outside of the brackets. 243. Uber divide not divided all to the exponent 4/5. There we go. Alright so negative. 243 over 32 to the exponent 4/5. Now we're gonna treat this like we did the first one. Okay we have an exponent of forfeits, we can treat this as four times 1/5. Our power rule allows us to break up that exponent and deal with it one thing at a time, so this is gonna be equal to negative, 243 divided by 32 to the exponent one fit all to the exponents for. Alright now careful where you put the negative. Okay it's good to include the brackets. Um Because if the negatives inside the brackets, the exponents going to apply to it when it's outside of the brackets, the exponent doesn't apply to it and that's going to change the sign of your answer. Okay so just be careful um with your negative sign that you're including it either outside the brackets or inside the brackets depending on where it should be. Alright so dealing with what's inside the brackett first, we know that we can apply this exponent 1/5 to the numerator and denominator. So we get 243 to the exponent 1/5 divided by 32 to the exponent 1/5. And this is all to the exponent for now we've done 243 to the exponent 1/5 and 32 to the exponent 1/5. When we did parts A. And B. So we know the values. Okay so we're gonna end up with 3/2. Okay all to the exponent four. Again we can apply the exponent to the numerator and denominator. So we have three to the exponent four divided by two to the exponent for and we end up with negative 81 over 16. Alright so for part B. Or for part C. Sorry are simplified expression worked out to be negative 81/16. So if we go back up to the top here expression C. We found to be negative 81/16. So that is gonna match with option B. And now we are on to our final expression. D. Okay we only have one option left in column B. So we expect the result to be negative 16/81. So let's work through it. Make sure that we didn't make any mistakes anywhere else. Alright so part D. R. Expression is negative. 243 divided by 32 to the exponent negative forfeits. Now we'll notice that if we look at this compared to part C, we have the same thing except the exponent is negative. Now when we did that with part A. And B. What we found was that the results for B. Where we had the negative exponent was just the inverse of what it was for party. Okay so that makes sense. We have negative 81/16. So we do probably expect the result to be negative 16/81. So between that being the only option left. Kind of looking at what we had for part C. That makes sense. So let's work it out. Alright so this is going to be again we can split that exponent up. We get 243 divided by 32 to the exponent 1/5. And all of this is going to be applied to an exponent of negative four now 243 divided by 32 to the exponent 1/5. We did it above in part C. And that worked out to be three halves. So this is gonna give us negative three halves to the exponent negative four. Okay now when we have an exponent of negative four we know that we can flip this and we can move it to the denominator. The entire base and the exponent and the exponent will become positive. Just be careful when you have a fraction K. The entire fraction has to move into that denominator not just one part of it. So we end up with negative one over three halves. The exponent four. We know that we can distribute our exponent to both the numerator and denominator of our base. Okay so we get three to the exponent four divided by two to the exponent four. Our rules for fractions tell us that this is negative two to the exponent four divided by three to the exponent four. Mm And using what we found in the other, true to the exponent four is gonna be 16. So we get negative 16 and three to the exponent four is 81. So we get negative 16/81 which is exactly what we expected. And so the simplified version of part D. Is negative 16/81. We go back up to our answer traces. We see that that means that D. Matches with D. Okay. All right now we need to choose and answer choice. So we have that A. Matches with C. So we're looking A matches with C. So we're looking at either option C. Or D. We found that B matches with A. C. Matches with B. And D. Matches with the So we are looking at answer C. Thanks everyone for watching. I hope this video helped see you in the next one.

Match each expression in Column I with its equivalent expression in Column II. See Example 8. a. (4/9)^3/2 b. (4/9)^-3/2 c. -(9/4)^3/2 d. -(9/4)^-3/2 A. 27/8 B. -27/8 C. 8/27 D. -8/27

Key Concepts

Exponents and Radicals

Negative Exponents

Simplifying Rational Expressions

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