In Exercises 117–124, simplify each exponential expression.

9y⁴/...

Question

In Exercises 117–124, simplify each exponential expression.

9y⁴/x⁻² + (x⁻¹/y²)⁻²

Accepted Answer

Hello, today we're gonna be simplifying the following expression. So what we are given is 13 Y to the power of eight over X to the power of negative four plus the quantity X to the power of negative two over Y to the power of four. And this quantity is race to the power of negative two. So let's go ahead and simplify the right hand side. Now, on the right hand side, we can go ahead and use the exponential rule of multiplication. And that states that if you have a to the power of M race to the power of N, you can just multiply the exponents together to give you a to the power of N times N. Here we're going to multiply the exponent of -2 to both the numerator and denominator. And that is going to give us X to the power of negative two times negative two over Y to the power of four times negative two. And this itself is going to simplify to give us X to the power of positive four over Y to the power of negative eight. Now we only want to express the answer using positive exponents. And if we use the, the property that allows us to take the reciprocal of a negative exponent, we can go ahead and write this as a positive exponent. Or in other words, if we take Y to the power of negative eight and move it to the numerator, then we can go ahead and write it as a positive exponent. So if we do this, we can rewrite X to the power of four over Y to the power of negative eight as X to the power of four times Y to the power of eight. And we're gonna do the same thing on the left hand side of the expression because in the denominator, we have X to the power of negative four. And if we take the reciprocal of this negative exponent by moving it to the numerator, then we can rewrite the left hand side as 13 X to the power of four times Y to the power of eight. So let's go ahead and combine the simplified forms of both sides of the expression together. So that is going to give us 13 X to the power of four Y to the power of eight plus X to the power of four Y to the power of eight. Since both of these exponential quantities are the same, we're just gonna go ahead and add the coefficient coefficients together. On the right hand side, this quantity has a coefficient of one and 13 plus one. Is going to give us 14. So the answer to this expression is going to be 14 X to the power of four Y to the power of eight. And this will be the simplified form of the given expression. And with that being said, the answer to this problem is going to be a. 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. 9y⁴/x⁻² + (x⁻¹/y²)⁻²

Key Concepts

Exponential Rules

Negative Exponents

Reciprocal and Power of a Fraction

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