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Algebra: Roots and Radicals

Read the following SAT test question and then select the correct answer.

Always read the question carefully and identify the bottom line.&nbsp. Assess your options for reaching the bottom line – what is the easiest and most time-efficient method to reach the answer? Use that method to attack the problem. When you have an answer, loop back to make sure that you reached the bottom line and did not just solve a portion of the problem.

If $\sqrt{x}=16$, what is the value of $\sqrt{4x}$?

Bottom Line: $\sqrt{4x}=?$

Assess your Options: You might be tempted to find the value of x first, but look at your bottom line. Do you need to know the value of x? No! Don’t waste your time! You just need to know the value of the square root of x multiplied by another number. Use your knowledge of radicals to rearrange your bottom line so that you have fewer steps to solve the problem.

Attack the Problem: Focus on the 4 under the radical. If this question simply asked for the square root of 4, you could easily answer. What is the square root of 4? 2! That value now goes in front of the radical. This could also be written as 2 multiplied by the square root of x. Plug in the value of 16 that you were given for the square root of x. All you have to do to reach a single number is multiply 2 by 16. Here are the steps you just completed:

$\sqrt{4x}= 2\sqrt{x}=(2)(\sqrt{x})=(2)(16)=32$

This method is much easier and faster than finding that x = 256, multiplying 256 by 4, and then taking the square root of 1024. You should not need to waste time typing numbers into a calculator in order to solve this problem.

Loop back: You solved for your bottom line, so you are ready to look at the answer choices.

$\sqrt{4x}= 2\sqrt{x}=(2)(\sqrt{x})=(2)(16)=32$

$\sqrt{4x}=?$

(A) 16

(B) 32

(C) 64

(D) 128

(E) 256

The correct answer is (B). On sat.collegeboard.org, 52% of the responses were correct.

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