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SAT Equivalent Expressions Question 23 | Free SAT Prep | Aniko

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Question 23·Medium·Equivalent Expressions

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Which of the following is equivalent to

(x^{3/2}y^{-6})^{2/3},

where $x>0$ and $y\neq 0$ ?

For exponent-equivalence questions, avoid expanding into radicals; instead, apply exponent rules directly and systematically. First, separate products raised to a power as $(ab)^n = a^n b^n$ , then use the power-of-a-power rule $(a^m)^n = a^{mn}$ to find new exponents. After simplifying those exponents (including fractions and negatives), rewrite any negative exponents using $a^{-n} = 1/a^n$ so the final expression uses only positive exponents, and then match that clean form to the answer choices. This approach is fast, reduces algebra mistakes, and works reliably for nested or fractional exponents on the SAT.

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Separate the factors inside the parentheses

Think of $(x^{3/2}y^{-6})^{2/3}$ as $(x^{3/2})^{2/3} \cdot (y^{-6})^{2/3}$ . What exponent rule lets you do that?

Use the power-of-a-power rule

For each factor, use $(a^m)^n = a^{mn}$ . Multiply the exponents on $x$ and on $y$ by $2/3$ separately.

Compute the new exponents carefully

Find $\frac{3}{2} \cdot \frac{2}{3}$ for the exponent on $x$ and $-6 \cdot \frac{2}{3}$ for the exponent on $y$ . Then think about how to rewrite a negative exponent using $a^{-n} = 1/a^n$ .

Match with the choices

Once you have $x$ raised to some exponent and $y$ raised to some (possibly negative) exponent, rewrite with only positive exponents and compare that form to the answer choices.

Desmos Guide

Enter the original expression with a fixed value of y

In Desmos, define a function using a convenient nonzero value for $y$ , such as $y=2$ . Type, for example, f(x) = (x^(3/2) * 2^(-6))^(2/3) (remember the domain is $x>0$ ).

Enter each answer choice as a separate function

Using the same fixed value of $y$ (e.g., $y=2$ ), enter:

g(x) = x^2 / 2^4
h(x) = x / 2^3
p(x) = x^(1/3) / 2^(-4)
q(x) = x / 2^4 These correspond to choices A, B, C, and D with $y$ replaced by $2$ .

Compare the graphs

Look at the graphs of $f(x)$ and each of the other functions for $x>0$ . The function whose graph exactly overlaps $f(x)$ for all viewed $x$ corresponds to the expression that is equivalent to the original.

Double-check with numeric values

Optionally, pick a few positive $x$ -values (like $x=1$ , $x=4$ , $x=9$ ), and use Desmos to evaluate $f(x)$ and each choice’s function at those $x$ -values. The correct expression will match $f(x)$ at every tested value.

Step-by-step Explanation

Identify the exponent rules you need

You are raising a product to a power: $(x^{3/2}y^{-6})^{2/3}$ . Two key rules apply:

Product to a power: $(ab)^n = a^n b^n$ .
Power to a power: $(a^m)^n = a^{mn}$ .

You will apply these rules separately to the $x$ part and the $y$ part.

Apply the power to each factor and multiply exponents

Rewrite the expression by applying the outside exponent $2/3$ to each factor:

(x^{3/2}y^{-6})^{2/3} = (x^{3/2})^{2/3} (y^{-6})^{2/3}.

Now use $(a^m)^n = a^{mn}$ :

For $x$ : the new exponent is

\frac{3}{2} \cdot \frac{2}{3}.

For $y$ : the new exponent is

-6 \cdot \frac{2}{3}.

Compute these products in the next step.

Simplify the exponents and handle the negative exponent

Calculate the exponents:

For $x$ :

\frac{3}{2} \cdot \frac{2}{3} = 1,

so you get $x^1$ .

For $y$ :

-6 \cdot \frac{2}{3} = -4,

so you get $y^{-4}$ .

Now the expression is

(x^{3/2}y^{-6})^{2/3} = x^1 y^{-4}.

Use the negative-exponent rule $a^{-n} = 1/a^n$ to rewrite $y^{-4}$ without a negative exponent.

Write the final simplified form and match the choice

Using $y^{-4} = 1/y^4$ and $x^1 = x$ , the simplified expression becomes

(x^{3/2}y^{-6})^{2/3} = \frac{x}{y^4}.

Comparing with the answer choices, this matches choice D, $\dfrac{x}{y^4}$ .

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation