Simplify: (1/x^7)(y^7) × Fourth Root of x^8

Let's begin by dealing with the root in the problem. We'll use the root and exponent law for this:

$\sqrt[n]{a^m}=(\sqrt[n]{a})^m=a^{\frac{m}{n}}$

Apply the above exponent law to the problem:

$\frac{1}{x^7}\cdot y^7\cdot\sqrt[4]{x^8}=\frac{1}{x^7}\cdot y^7\cdot x^{\frac{8}{4}}=\frac{1}{x^7}\cdot y^7\cdot x^2$

When in the first stage we applied the above law to the third term in the product. We did this carefully whilst paying attention to what goes into the numerator of the fraction in the exponent. Let's ask ourselves what goes into the denominator of the fraction in the exponent? In the following stages, we simplified the expression that we obtained.

Next, we'll recall the exponent law for negative exponents in the opposite direction:

$\frac{1}{a^n} =a^{-n}$

We'll apply this exponent law to the first term in the product in the expression that we obtained in the last stage:

$\frac{1}{x^7}\cdot y^7\cdot x^2=x^{-7}\cdot y^7\cdot x^2=y^7\cdot x^{-7}\cdot x^2$

When in the first stage we applied the above exponent law to the first term in the product and in the next stage we arranged the expression that we obtained by using the commutative property of multiplication. Hence terms with identical bases are adjacent to each other.

Next, we'll recall the exponent law for multiplying terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

Apply this exponent law to the expression that we obtained in the last stage:

$y^7\cdot x^{-7}\cdot x^2=y^7x^{-7+2}=y^7x^{-5}$

When in the first stage we applied the above exponent law for the terms with identical bases, and then proceeded to simplify the expression that we obtained. Additionally in the final stages we removed the · sign and switched to the conventional notation where placing terms next to each other signifies multiplication.

Let's summarize the various steps of the solution so far:

$\frac{1}{x^7}\cdot y^7\cdot\sqrt[4]{x^8}=\frac{1}{x^7}\cdot y^7\cdot x^{\frac{8}{4}}=x^{-7}y^7x^2=y^7x^{-5}$

Therefore, the correct answer is answer D.