Easy Algebra Step-by-Step: Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST! (2012)
Chapter 3. Roots and Radicals
In this chapter, you learn about square roots, cube roots, and so on. Additionally, you learn about radicals and their relationship to roots. It is important in algebra that you have a facility for working with roots and radicals.
Squares, Square Roots, and Perfect Squares
You square a number by multiplying the number by itself. For instance, the square of 4 is 4 · 4 = 16. Also, the square of –4 is –4 · –4 = 16. Thus, 16 is the result of squaring 4 or –4. The reverse of squaring is finding the square root. The two square roots of 16 are 4 and –4. You use the symbol 
to represent the positive square root of 16. Thus, 
. This number is the principal square root of 16. Thus, the principal square root of 16 is 4. Using the square root notation, you indicate the negative square root of 16 as 
. Thus, 
.
is not a real number because no real number multiplies by itself to give –16.
Every positive number has two square roots that are equal in absolute value, but opposite in sign. The positive square root is called the principal square root of the number. The number 0 has only one square root, namely, 0. The principal square root of 0 is 0. In general, if x is a real number such that x · x = s then 
(the absolute value of x).
The 
symbol always gives one number as the answer and that number is nonnegative: positive or 0.
A number that is an exact square of another number is a perfect square. For instance, the integers 4, 9, 16, and 25 are perfect squares. Here is a helpful list of principal square roots of some perfect squares.
Working with square roots will be much easier for you if you memorize the list of square roots. Make flashcards to help you do this.
Also, fractions and decimals can be perfect squares. For instance, 
is a perfect square because 
equals 
, and 0.36 is a perfect square because 0.36 equals (0.6)(0.6). If a number is not a perfect square, you can indicate its square roots by using the square root symbol. For instance, the two square roots of 15 are 
and –.
Problem Find the two square roots of the given number.
a. 25
b. 
c. 0.49
d. 11
Solution
a. 25
Step 1. Find the principal square root of 25.
5 · 5 = 25, so 5 is the principal square root of 25.
Step 2. Write the two square roots of 25.
5 and –5 are the two square roots of 25.
b. 
Step 1. Find the principal square root of .
, so 
is the principal square root of .
Step 2. Write the two square roots of .
and 
are the two square roots of .
c. 0.49
Step 1. Find the principal square root of 0.49.
(0.7)(0.7) = 0.49, so 0.7 is the principal square root of 0.49.
Step 2. Write the two square roots of 0.49.
0.7 and –0.7 are the two square roots of 0.49.
d. 11
Step 1. Find the principal square root of 11.
is the principal square root of 11.
Step 2. Write the two square roots of 11.
and 
are the two square roots of 11.
Because 11 is not a perfect square, you indicate the square root.
Problem Find the indicated root.
a. 
b. 
c. 
d. 
e. 
f. 
g. 
Solution
a. 
Step 1. Find the principal square root of 81.
. The square root symbol always gives just one nonnegative number as the answer! If you want ±9, then do this: 
.
b. 
Step 1. Find the principal square root of 100.
. You do not divide by 2 to get a square root.
c. 
Step 1. Find the principal square root of 
.
d. 
Step 1. Find the principal square root of 30.
Because 30 is not a perfect square, 
indicates the principal square root of 30.
e. 
Step 1. Add 9 and 16 because you want the principal square root of the quantity 9 + 16. (See Chapter 5 for a discussion of 
as a grouping symbol.)
Step 2. Find the principal square root of 25.
, but 
.
f. 
Step 1. Find the principal square root of –2 · –2.
. The 
symbol never gives a negative number as an answer.
g. 
Step 1. Find the principal square root of b · b.
if b is negative and |b| ≠ b if b is negative. Because you don’t know the value of the number b, you must keep the absolute value bars.
Cube Roots and nth Roots
A number x such that x · x · x = c is a cube root of c. Finding the cube root of a number is the reverse of cubing a number. Every real number has exactly one real cube root, called its principal cube root. For example, because –4 · –4 · –4 = –64, –4 is the principal cube root of –64. You use 
to indicate the principal cube root of –64. Thus, 
. Similarly, 
. As you can see, the principal cube root of a negative number is negative, and the principal cube root of a positive number is positive. In general, if x is a real number such that x · x · x = c, then 
. Here is a list of principal cube roots of some perfect cubes that are useful to know.
You will find it worth your while to memorize the list of cube roots.
If a number is not a perfect cube, you indicate its principal cube root by using the cube root symbol. For instance, the cube root of –18 is 
.
Problem Find the indicated root.
a. 
b. 
c. 
d. 
e. 
f. 
Solution
a. 
Step 1. Find the principal cube root of –27.
.
. You do not divide by 3 to get a cube root.
b. 
Step 1. Find the principal cube root of 
.
.
c. 
Step 1. Find the principal cube root of 0.008.
d. 
Step 1. Find the principal cube root of –1.
–1 · –1 · –1 = –1, so 
.
e. 
Step 1. Find the principal cube root of –7 · –7 · –7.
f. 
Step 1. Find the principal cube root of b · b · b.
In general, if 
, where n is a natural number, x is called an nth root of a. The principal nth root of a is denoted 
. The expression 
is called a radical, a is called the radicand, n is called the index and indicates which root is desired. If no index is written, it is understood to be 2 and the radical expression indicates the principal square root of the radicand. As a rule, a positive real number has exactly one real positive nth root whether n is even or odd, and every real number has exactly one real nth root when n is odd. Negative numbers do not have real nth roots when n is even. Finally, the nth root of 0 is 0 whether n is even or odd: 
(always).
Problem Find the indicated root, if possible.
a. 
b. 
c. 
d. 
e. 
f. 
Solution
a. 
Step 1. Find the principal fourth root of 81.
.
b. 
Step 1. Find the principal fifth root of 
.
.
c. 
Step 1. Find the principal cube root of 0.125.
.
d. 
Step 1. –1 is negative and 6 is even, so 
is not a real number.
is not defined for real numbers.
, not –1.
e. 
Step 1. Find the principal seventh root of –1.
f. 
Step 1. Find the principal 50th root of 0.
The nth root of 0 is 0, so 
.
Simplifying Radicals
Sometimes in algebra you have to simplify radicals—most frequently, square root radicals. A square root radical is in simplest form when it has (a) no factors that are perfect squares and (b) no fractions. You use the following property of square root radicals to accomplish the simplifying.
If a and b are nonnegative numbers,
Problem Simplify.
a. 
b. 
c. 
d. 
Solution
a. 
Step 1. Express 
as a product of two numbers, one of which is the largest perfect square.
Step 2. Replace 
with the product of the square roots of 16 and 3.
Step 3. Find 
and put the answer in front of 
as a coefficient. (See Chapter 6 for a discussion of the term coefficient.)
Step 4. Review the main results.
b. 
Step 1. Express 
as a product of two numbers, one of which is the largest perfect square.
Step 2. Replace 
with the product of the square roots of 36 and 10.
Step 3. Find 
and put the answer in front of 
as a coefficient.
Step 4. Review the main results.
c. 
Step 1. Express as a product of two numbers, one of which is the largest perfect square.
Step 2. Replace 
with the product of the square roots of 
and 3.
Step 3. Find 
and put the answer in front of 
as a coefficient.
Step 4. Review the main results.
d. 
Step 1. Multiply the numerator and the denominator of 
by the least number that will make the denominator a perfect square.
Step 2. Express 
as a product of two numbers, one of which is the largest perfect square.
Step 3. Replace 
with the product of the square roots of 
and 2.
Step 4. Find 
and put the answer in front of 
as a coefficient.
Step 5. Review the main results.
Exercise 3
For 1–4, find the two square roots of the given number.
1. 144
2. 
3. 0.64
4. 400
For 5–18, find the indicated root, if possible.
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
For 19 and 20, simplify.
19. 
20. 