SAT Test Prep


Lesson 4: Working with Roots

What Are Roots?

The Latin word radix means root (remember that radishes grow underground), so the word radical means the root of a number (or a person who seeks to change a system “from the roots up”). What does the root of a plant have to do with the root of a number?

Think of a square with an area of 9 square units sitting on the ground:

The bottom of the square is “rooted” to the ground, and it has a length of 3. So we say that 3 is the square root of 9!

The square root of a number is what you must square to get the number.

All positive numbers have two square roots. For instance, the square roots of 9 are 3 and –3.

The radical symbol, , however, means only the non-negative square root. So although the square root of 9 equals either 3 or –3,  equals only 3.

The number inside a radical is called a radicand.


If x2 is equal to 9 or 16, then what is the least possible value of x3?

x is the square root of 9 or 16, so it could be –3, 3, –4, or 4. Therefore, x3 could be –27, 27, –64, or 64. The least of these, of course, is –64.

Remember that  does not always equal x. It does, however, always equal |x|.



Don’t worry about squaring first, just remember the rule above. It simplifies to


Working with Roots

Memorize the list of perfect squares: 4, 9, 16, 25, 36, 49, 64, 81, 100. This will make working with roots easier.

To simplify a square root expression, factor any perfect squares from the radicand and simplify.


When adding or subtracting roots, treat them like exponentials: combine only like terms—those with the same radicand.


When multiplying or dividing roots, multiply or divide the coefficients and radicands separately.


You can also use the commutative and associative laws when simplifying expressions with radicals.


Concept Review 4: Working with Roots

1. List the first 10 perfect square integers greater than 1: _________________________________________________

2. How can you tell whether two radicals are “like” terms?

3. An exponential is a perfect square only if its coefficient is _____ and its exponent is _____.

For questions 4–7, state whether each equation is true (T) or false (F). If it is false, rewrite the expression on the right side to correct it.

Simplify the following expressions, if possible.

SAT Practice 4: Working with Roots

1. The square root of a certain positive number is twice the number itself. What is the number?

(E) 1

2. If , what is one possible value of x?

3. If  and , what is the greatest possible value of a – b?

(A) –3

(B) –1

(C) 3

(D) 5

(E) 7

4. If , then 

(E) 18

5. If ,, and , then 

(A) –35

(B) –19

(C) 0

(D) 19

(E) 35

6. If m and n are both positive, then which of the following is equivalent to ?

7. A rectangle has sides of length cm and cm. What is the length of a diagonal of the rectangle?

8. The area of square A is 10 times the area of square B. What is the ratio of the perimeter of square A to the perimeter of square B?

9. In the figure above, if n is a real number greater than 1, what is the value of x in terms of n?

Answer Key 4: Working with Roots

Concept Review 4

1. 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 2.

2. They are “like” if their radicands (what’s inside the radical) are the same.

3. An exponential is a perfect square only if its coefficient is a perfect square and its exponent is even.

4. false:

5. true:

6. true:

7. false if x is negative:

8. 5 or –5

9. 64 or –64

 (Law of Distribution)

 can’t be simplified (unlike terms).

SAT Practice 4

1. B The square root of ¼ is ½, because . Twice ¼ is also ½, because . You can also set it up algebraically:

2. Any number between 1 and 4 (but not 1 or 4). Guess and check is probably the most efficient method here. Notice that  only if , and  only if .

3. E , so  or –3. , so  or –4. The greatest value of , then, is .

5. D If , then , and if , then . But if , then x cannot be 2 and y cannot be –3. Therefore,  and y = 3. 

Also, you can plug in easy positive values for m and n like 1 and 2, evaluate the expression on your calculator, and check it against the choices.

7. C The diagonal is the hypotenuse of a right triangle, so we can find its length with the Pythagorean theorem:

Or you can plug in numbers for a and b, like 9 and 16, before you use the Pythagorean theorem.

8. C Assume that the squares have areas of 10 and 1. The lengths of their sides, then, are  and 1, respectively, and the perimeters are  and 4.

9. B Use the Pythagorean theorem:

Take the square root:  (Or plug in!)