In Exercises 117–124, simplify each exponential expression.(3x⁴/y...

Question

In Exercises 117–124, simplify each exponential expression.(3x⁴/y⁻⁴)⁻¹(2x/y²)³

Accepted Answer

Hello, today we're gonna be simplifying the given expression. So what we are given is the quantity five times X to the power of six over Y to the power of negative two. That quantity raced to the negative first, power times the quantity six X squared over Y to the fifth race to the power of two. Now, the first thing we're going to do is we're going to use the exponential multiplication to simplify the outer exponents. And that states that if you have the quantity A to the power of M race, the power of N, then you can just multiply the exponents together to give you A to the power of M times N. Here, on the left hand side, we're going to distribute the power of negative 1 to 5 and X to the sixth. And we're also going to distribute the negative first power to Y to the power of negative two. And we're going to do the same thing. On the right hand side, we're going to distribute the exponent of 2 to 6 and X squared and we're going to distribute it to Y to the power of fifth. So on the left-hand side. When we distribute this, we get five to the power of negative one times X to the power of six times negative one. And that is going to be over Y to the power of negative two times negative one. And on the right hand side, distributing the exponent is going to give us six to the power of two times X to the power of two times two over Y to the power of 5 times 2. Now we're going to simplify each of the exponents on the left hand side, six times negative one is going to give us negative six. So we can rewrite this to give us five to the power of negative one times X to the power of negative six. And in the denominator negative two times negative one is going to give us positive two. So we have Y to the power of positive two in the denominator. And on the right hand side, X to the power of two times two is going to give us positive four. So we have X to the power of four on the right hand side. And in the denominator on the right hand side, five times two is going to give us 10. So we have Y to the power of positive 10. Now we can go ahead and multiply the two fractions together and we're gonna multiply numerator with numerator and denominator with denominator. And that is going to give us five to the power of negative one times six squared times X to the power of negative six times X to the fourth over Y squared times Y to the 10th. Now, here we can go ahead and simplify the numerator and denominator by using our exponential edition. And that states that if you have a, to the power of M times A to the power of N, you can combine these two exponential quantities together by adding the exponents together. And this can be rewritten as A to the power of N plus N. So when the denominator, I mean in the enumerator X to the power of negative six times X to the power of four can be rewritten as X to the power of negative six plus four. And this is going to simplify to give us X to the power of negative two. Furthermore, in the numerator six squared is the same thing as six times six, which is going to give us 36. So we can rewrite the numerator as five to the power of negative one times 36 times X to the power of negative two. And in the denominator, if we use the exponential addition, again, we can rewrite Y squared times Y to the power of 10 as Y to the power of two plus 10. And this is going to give us Y to the power of 12. So the denominator simplifies to give us Y to the power of 12. Finally, we want to express the answer as using only positive exponents. So in the numerator, we have five to the power of negative one and X to the power of negative two. If we take the reciprocal of these two values, by moving it to the denominator, we can rewrite them using positive exponents. And this expression can be rewritten as 36/5 to the power of one, which is just going to give us five times X to the power of positive two times Y to the power of 12. And this is going to be the simplified form of the given expression. And with that being said, the answer to this problem is going to be B so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 117–124, simplify each exponential expression.(3x⁴/y⁻⁴)⁻¹(2x/y²)³

Key Concepts

Exponential Rules

Negative Exponents

Combining Exponential Expressions

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