Let and be positive real numbers. Which of the following expressions is equivalent to the expression above?

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SAT Equivalent Expressions Question 227 | Free SAT Prep | Aniko

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Question 227·Medium·Equivalent Expressions

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Let $x$ and $y$ be positive real numbers.

x^{3/2} y^{-1/2} \cdot \sqrt{\frac{y}{x}}

Which of the following expressions is equivalent to the expression above?

For exponent and radical simplification questions, first rewrite all roots as fractional exponents, then express everything with separate powers of each base (like $x$ and $y$ ). Use exponent rules ( $a^m a^n = a^{m+n}$ ) to combine like bases, watch signs carefully for negative exponents, and remember that a zero exponent gives $1$ . Once the expression is fully simplified, match it to the answer choice instead of trying to work backward from the choices.

Hints

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Turn the square root into an exponent

Rewrite $\sqrt{\dfrac{y}{x}}$ using a fractional exponent. What power corresponds to a square root?

Separate the fraction inside the square root

After writing $\sqrt{\dfrac{y}{x}}$ as a power, split it into a product involving a power of $y$ and a power of $x$ .

Combine like bases using exponent rules

Group all the $x$ terms together and all the $y$ terms together. Add exponents for the same base, and remember what happens when an exponent is $0$ .

Desmos Guide

Assign numerical values to x and y

Pick any convenient positive numbers (for example, type x = 4 on one line and y = 9 on another). Using positive values is important because of the square root.

Evaluate the original expression

On a new line, type the original expression exactly: x^(3/2) * y^(-1/2) * sqrt(y/x). Note the numerical value Desmos gives for this expression.

Test each answer choice numerically

On separate lines, type each option using the same x and y values: x, y, x*y, and sqrt(x*y). Compare their numerical values to the value from the original expression. The option whose value matches the original expression shows the correct equivalent expression.

Step-by-step Explanation

Write the expression and note the goal

We want to simplify

x^{3/2} y^{-1/2} \cdot \sqrt{\frac{y}{x}}

and then see which answer choice matches the simplified form.

Rewrite the square root using exponents

A square root can be written as a power of $1/2$ .

\sqrt{\frac{y}{x}} = \left(\frac{y}{x}\right)^{1/2}

Now split this fraction inside the exponent:

\left(\frac{y}{x}\right)^{1/2} = y^{1/2} x^{-1/2}

So the whole expression becomes

x^{3/2} y^{-1/2} \cdot y^{1/2} x^{-1/2}.

Group like bases and combine exponents

Now group the $x$ terms together and the $y$ terms together.

For the $x$ factors:

x^{3/2} \cdot x^{-1/2} = x^{3/2 + (-1/2)} = x^{(3/2 - 1/2)} = x^{2/2} = x^1.

For the $y$ factors:

y^{-1/2} \cdot y^{1/2} = y^{-1/2 + 1/2} = y^0.

So the expression simplifies to

x^1 \cdot y^0.

Use the meaning of exponent 0 and conclude

Any nonzero number to the zero power is $1$ , so $y^0 = 1$ .

Therefore,

x^1 \cdot y^0 = x \cdot 1 = x.

So the expression is equivalent to $x$ , which matches choice A.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation