For and , which of the following expressions is equivalent to

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SAT Equivalent Expressions Question 209 | Free SAT Prep | Aniko

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Question 209·Medium·Equivalent Expressions

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For $a>0$ and $b>0$ , which of the following expressions is equivalent to

\frac{\sqrt{49a^3b^5}}{ab^{2}}

For radical expressions with variables, first factor the expression under the square root into perfect squares and leftovers so you can pull as many factors as possible out of the radical using $\sqrt{x^2} = x$ (for $x>0$ ). Then write the entire numerator as a product of ordinary factors and a single square root, and cancel any common factors with the denominator. Avoid the extra step of moving denominators into the radical unless you are very comfortable with squaring them—simplifying the radical first and then reducing the fraction is usually faster and less error-prone on the SAT.

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Separate the radical

Try rewriting $\sqrt{49a^3b^5}$ as a product of square roots, like $\sqrt{49}\,\sqrt{a^3}\,\sqrt{b^5}$ . Which parts inside each factor are perfect squares?

Use perfect squares to pull factors out

Remember that $\sqrt{x^2} = x$ for $x>0$ . Can you express $a^3$ and $b^5$ as (square) $\times$ (leftover) so that you can take some $a$ 's and $b$ 's out of the radical?

Simplify the fraction

Once you have written $\sqrt{49a^3b^5}$ as something like (number) $\times a^m b^n \times$ (square root), look for common $a$ and $b$ factors in both the numerator and the denominator $ab^2$ that can be canceled.

Avoid moving the denominator inside the square root

It is usually easier (and safer) to simplify the radical in the numerator first, then divide by $ab^2$ , instead of trying to combine everything under one square root at the start.

Desmos Guide

Define positive values for a and b

In Desmos, type a=4 and b=9 (or choose any positive numbers, since $a>0$ and $b>0$ ). You can also make a and b sliders if you like by clicking the slider icons.

Enter the original expression

In a new line, type the original expression as expr = sqrt(49*a^3*b^5)/(a*b^2). Note the numerical value that Desmos gives for expr for your chosen $a$ and $b$ .

Enter each answer choice as a separate expression

On new lines, type each option: A = 7*a*sqrt(b), B = 7*sqrt(a/b), C = 7*sqrt(a*b), D = 7/sqrt(a*b). For your chosen $a$ and $b$ , compare the values of A, B, C, and D to the value of expr and see which one matches.

Verify with different values

Change the values of a and b (either by editing them or using sliders) to other positive numbers and check again which option always produces the same value as expr. The expression that consistently matches expr for all tested positive $a$ and $b$ is the correct equivalent form.

Step-by-step Explanation

Factor perfect squares inside the square root

Start with

\frac{\sqrt{49a^3b^5}}{ab^{2}}.

Break the expression under the radical into perfect-square parts and leftovers:

$49 = 7^2$
$a^3 = a^2 \cdot a$
$b^5 = b^4 \cdot b$

\sqrt{49a^3b^5} = \sqrt{7^2}\,\sqrt{a^2}\,\sqrt{a}\,\sqrt{b^4}\,\sqrt{b} = 7ab^2\sqrt{ab}.

Now the whole expression becomes

\frac{\sqrt{49a^3b^5}}{ab^2} = \frac{7ab^2\sqrt{ab}}{ab^2}.

Cancel common factors with the denominator

\frac{7ab^2\sqrt{ab}}{ab^2},

$a$ in the numerator cancels with $a$ in the denominator, and $b^2$ in the numerator cancels with $b^2$ in the denominator. Because $a>0$ and $b>0$ , we do not need absolute values when taking square roots.

After canceling, only the constant and the square root factor remain.

Write the simplified expression and match the choice

After canceling $a$ and $b^2$ , we get

\frac{7ab^2\sqrt{ab}}{ab^2} = 7\sqrt{ab}.

So the original expression is equivalent to $7\sqrt{ab}$ , which corresponds to choice C.

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Step-by-step Explanation

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