After you simplify the numerator, you will have a fraction with powers of $a$ and powers of $b$ on top and bottom. Use the rule that $\dfrac{x^m}{x^n} = x^{m-n}$ when $x$ is the same base.

Once you have a single power of $a$ and a single power of $b$, rewrite any negative exponent using the idea that a negative exponent means a reciprocal (for example, $x^{-k} = 1/x^{k}$).

Make sure your final expression has only positive exponents and then match it to the answer choice that has the same powers of $a$ and $b$ and the same numerical coefficient.

In Desmos, define positive values for the variables, such as a = 2 and b = 3. Any positive values will work because the expression is defined for $a>0$ and $b>0$.

On a new line, type the original expression exactly as given, for example: E1 = (2a^(-3)*b^4)^2 / (8*a^(-1)*b) Note the use of ^(-3) and ^4 for exponents.

On new lines, enter each option using the same values of $a$ and $b$, for example:

• A = 2*b^7 / a^5
• B = a^5 / (2*b^7)
• C = b^8 / (4*a^6)
• D = b^7 / (2*a^5) Desmos will display a numerical value for each.

Compare the numerical value of E1 with the values of A, B, C, and D. The choice whose value matches E1 for your chosen positive $a$ and $b$ is the expression equivalent to the original.

Start with the numerator:

Apply the rule $(xy)^2 = x^2y^2$ and the power rule $(x^m)^2 = x^{2m}$:

So the whole expression becomes

Rewrite the fraction by grouping the numerical coefficients, the $a$-terms, and the $b$-terms:

Now you can simplify each part separately.

First, simplify the numerical fraction:

Next, use the quotient rule for exponents: when you divide powers with the same base, subtract exponents.

• For the $a$-terms:

• For the $b$-terms:

So the expression is now

A negative exponent means a reciprocal: $a^{-5} = \dfrac{1}{a^{5}}$. Use this to rewrite the expression:

Comparing with the answer choices, this matches choice D) $\dfrac{b^{7}}{2a^{5}}$.