After simplifying the numerator, you will have something of the form $\dfrac{2x^{a}y^{b}}{2x^{-1}y^{3/2}}$. Cancel the $2$ and then use $a^m/a^n = a^{m-n}$ to combine the exponents on $x$ and on $y$.

Once you have simplified the exponents, if any variable still has a negative exponent, rewrite it as a positive exponent in the denominator using $a^{-k} = 1/a^k$.

In Desmos, define two sliders or constants for positive values, for example type a = 4 and b = 2, so you can substitute $x = a$ and $y = b$ into the expressions.

On a new line, type the original expression using a and b:

((16*a^6*b^-2)^(1/4))/(2*a^-1*b^(3/2))

Note the numeric value that Desmos outputs for this expression.

For each option A–D, type the corresponding expression with a and b (for example, a^(5/2)/b^2 for one choice). Compare each output to the value from the original expression. The choice that always matches the original value (and still works if you change a and b to other positive numbers) is the correct equivalent expression.

Start with the expression:

Apply the rule $(abc)^n = a^n b^n c^n$ to the numerator:

Now apply $(a^m)^n = a^{mn}$ to each power term in the numerator, keeping them in exponent form for now.

Evaluate each factor in the numerator:

• $16^{1/4} = 2$ because $2^4 = 16$.
• $(x^{6})^{1/4} = x^{6 \cdot \frac{1}{4}} = x^{3/2}$.
• $(y^{-2})^{1/4} = y^{-2 \cdot \frac{1}{4}} = y^{-1/2}$.

So the whole expression becomes

Now cancel the common factor of $2$ in numerator and denominator, and use the rule for dividing powers with the same base, $a^m / a^n = a^{m-n}$, to combine the $x$ and $y$ terms:

Simplify the exponents:

• For $x$: $3/2 - (-1) = 3/2 + 1 = 3/2 + 2/2 = 5/2$, so we get $x^{5/2}$.
• For $y$: $-1/2 - 3/2 = -4/2 = -2$, so we get $y^{-2}$.

Thus the expression becomes

Rewrite the negative exponent using $a^{-k} = 1/a^k$:

So the expression is equivalent to $\dfrac{x^{5/2}}{y^2}$.