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Roots and Radicals

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 , what is the value of ?

Bottom Line:

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:



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.





(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.

For more help with SAT math, visit www.myknowsys.com!

Roots and Radicals

Algebra: Roots and Radicals

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

Always read the problem carefully and determine the bottom line, the question that you must answer.  Assess your options for solving the problem and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that you completed all the necessary steps and solved for the bottom line.

If  , which of the following must be true?

Bottom Line: Which of the following . . . ?

Assess your Options:  Many "Which of the following . . . " questions require you to look at the answer choices to solve the problem, but you should always check to see whether you can simplify the equation that you have been given.  Instead of jumping to the answer choices, work the equation into a form that is not as intimidating.

Attack the Problem:  The original equation has a square root on each side.  How do you get rid of these square root signs?  Square both sides of the equation, and the roots will cancel out.  You are left with:

xa = x + b

You just showed that when something is on both sides of the equation, you can cancel it out.  There is a positive x on both sides of the equation.  If you subtract it from one side, you must subtract it from the other, and the x is eliminated.  You are left with:

-a = b

This looks fairly simple, so glance down at your answer choices.  All of them are set equal to 0.  Set your equation equal to zero by adding an a to each side.

0 = b + a

Remember, it doesn’t matter what order you use when adding two variables. 

Loop Back:  You put your answer in the same form as the answers on the test, so now all you have to do is match your answer to the correct one!

(A) a = 0
(B) b = 0
(C) a + b = 0
(D) a b = 0
(E) a² + b² = 0

The correct answer is (C).


On sat.collegeboard.org, 54% of the responses were correct.

For more help with SAT vocabulary, visit www.myknowsys.com!

Roots and Radicals

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

Read the question carefully to insure that you understand all of the information that you are given.  Then make a note of the bottom line so that you will be sure to solve for the correct information.  Assess your options for solving the problem and choose the most efficient method.  Attack the problem, solve it, and loop back to verify that your answer matches the bottom line.

If  , what is the value of ?

Bottom Line: You must find what  is equal to.

Assess your Options:  You could solve for x.  However, that is actually the long way to do this problem.  You are taking a timed test, so the long way is the wrong way!  Your bottom line is , so you can use what you know about radicals to solve the problem without solving for x.

Attack the Problem:  Look first at .  This is actually the same as  times 
.  Finding the square root of 4 is easy: .  Now you have .  Look back at the problem.  You are already given the square root of x!.  What is 2 times 16?  The answer is 32.  This is the process that you just followed:

Loop Back: Did you find the bottom line? Yes; .  Look down at your answer choices.

(A) 16
(B) 32
(C) 64
(D) 128
(E) 256

The correct answer is (B).


On sat.collegeboard.org, 36% of the responses were correct.

For more help with math, visit www.myknowsys.com!

Roots and Radicals

Link of the Day

Every year 1.2 million students drop out. That's 857 students for every hour of every school day. In an effort to bring attention to this, College Board (the company that produces and administers the SAT) has set up 857 empty desks at the National Mall in Washington D.C. You can read more about College Board's "Don't Forget Ed" campaign here. This would make a great Excellent Example for your essay.

6/20 Algebra: Roots and Radicals

Read the following SAT question and then select your answer.

If square root (x minus a) = square root (x + b), which of the following must be true?

Step 1 of the Knowsys method for math is always to read the question carefully. Then, identify the bottom line. In this case, you need to select the answer choice that is true given the formula above. Step 3 is to evaluate your options. Think about what you could do, and what you should do. Since you have variables in the problem and in the answer choices, you could pick numbers for two of the variables, solve the equation for the third variable, and then look at the various answer choices. However, since this problem looks fairly straightforward, that's not necessary. It is faster and easier to simply manipulate and simplify the equation and then look at the answer choices (so this is what you should do). Note that if you did get stuck in the simplifying process, you could always go back and try the other method.

Start by squaring both sides of the equation.



Now, simply subtract x from both sides of the equation.



Lastly, add a to both sides of the equation.



Now, you know that a is equal to -b and you also know that the value of x doesn't matter (since it was eliminated). Take a look at the answer choices and see which one must be true. Don't forget to check answer choice (E) first and then work backwards. On "which of the following" questions, the test makers know that you will probably start with answer choice (A) and work your way down. Because they want you to as much time as possible, they usually put the correct answer near the end (not always, but usually).











Answer choice (E) would only be true if both a and b were equal to 0. Since you are looking for the answer choice that must be true, (E) won't work. You know that (D) is incorrect because it does not match the simplified form of the equation. (C) matches your prediction exactly. (B) and (A) both could be true, but they do not have to be true so neither of them is the correct answer choice.

The correct answer choice is (C).

On sat.collegeboard.org 57% of the responses were correct.

For more help with math, visit www.myknowsys.com.