Roots and Radicals - Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST - Easy Algebra Step-by-Step

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 Image to represent the positive square root of 16. Thus, Image. 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 Image. Thus, Image.


Image 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 Image (the absolute value of x).


The Image 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.

Image


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, Image is a perfect square because Image equals Image, 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 Image and –Image.

Problem Find the two square roots of the given number.

a. 25

b. Image

c. 0.49

d. 11

Solution

a. 25

Image 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. Image

Image Step 1. Find the principal square root of Image.

Image, so Image is the principal square root of Image.

Step 2. Write the two square roots of Image.

Image and Image are the two square roots of Image.

c. 0.49

Image 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

Image Step 1. Find the principal square root of 11.

Image is the principal square root of 11.

Step 2. Write the two square roots of 11.

Image and Image 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. Image

b. Image

c. Image

d. Image

e. Image

f. Image

g. Image

Solution

a. Image

Image Step 1. Find the principal square root of 81.

Image


Image. The square root symbol always gives just one nonnegative number as the answer! If you want ±9, then do this: Image.


b. Image

Image Step 1. Find the principal square root of 100.

Image


Image. You do not divide by 2 to get a square root.


c. Image

Image Step 1. Find the principal square root of Image.

Image

d. Image

Image Step 1. Find the principal square root of 30.

Because 30 is not a perfect square, Image indicates the principal square root of 30.

e. Image

Image 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 Image as a grouping symbol.)

Image

Step 2. Find the principal square root of 25.

Image


Image, but Image.


f. Image

Image Step 1. Find the principal square root of –2 · –2.

Image


Image. The Image symbol never gives a negative number as an answer.


g. Image

Image Step 1. Find the principal square root of b · b.

Image


Image 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 Image to indicate the principal cube root of –64. Thus, Image. Similarly, Image. 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 Image. Here is a list of principal cube roots of some perfect cubes that are useful to know.

Image


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 Image.

Problem Find the indicated root.

a. Image

b. Image

c. Image

d. Image

e. Image

f. Image

Solution

a. Image

Image Step 1. Find the principal cube root of –27.

Image.


Image. You do not divide by 3 to get a cube root.


b. Image

Image Step 1. Find the principal cube root of Image.

Image.

c. Image

Image Step 1. Find the principal cube root of 0.008.

Image

d. Image

Image Step 1. Find the principal cube root of –1.

–1 · –1 · –1 = –1, so Image.

e. Image

Image Step 1. Find the principal cube root of –7 · –7 · –7.

Image

f. Image

Image Step 1. Find the principal cube root of b · b · b.

Image

In general, if Image, where n is a natural number, x is called an nth root of a. The principal nth root of a is denoted Image. The expression Image 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: Image (always).

Problem Find the indicated root, if possible.

a. Image

b. Image

c. Image

d. Image

e. Image

f. Image

Solution

a. Image

Image Step 1. Find the principal fourth root of 81.

Image.

b. Image

Image Step 1. Find the principal fifth root of Image.

Image.

c. Image

Image Step 1. Find the principal cube root of 0.125.

Image.

d. Image

Image Step 1. –1 is negative and 6 is even, so Image is not a real number.

Image is not defined for real numbers.


Image, not –1.


e. Image

Image Step 1. Find the principal seventh root of –1.

Image

f. Image

Image Step 1. Find the principal 50th root of 0.

The nth root of 0 is 0, so Image.

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.

Image If a and b are nonnegative numbers,

Image

Problem Simplify.

a. Image

b. Image

c. Image

d. Image

Solution

a. Image

Image Step 1. Express Image as a product of two numbers, one of which is the largest perfect square.

Image

Step 2. Replace Image with the product of the square roots of 16 and 3.

Image

Step 3. Find Image and put the answer in front of Image as a coefficient. (See Chapter 6 for a discussion of the term coefficient.)

Image

Step 4. Review the main results.

Image

b. Image

Image Step 1. Express Image as a product of two numbers, one of which is the largest perfect square.

Image

Step 2. Replace Image with the product of the square roots of 36 and 10.

Image

Step 3. Find Image and put the answer in front of Image as a coefficient.

Image

Step 4. Review the main results.

Image

c. Image

Image Step 1. Express Image as a product of two numbers, one of which is the largest perfect square.

Image

Step 2. Replace Image with the product of the square roots of Image and 3.

Image

Step 3. Find Image and put the answer in front of Image as a coefficient.

Image

Step 4. Review the main results.

Image

d. Image

Image Step 1. Multiply the numerator and the denominator of Image by the least number that will make the denominator a perfect square.

Image

Step 2. Express Image as a product of two numbers, one of which is the largest perfect square.

Image

Step 3. Replace Image with the product of the square roots of Image and 2.

Image

Step 4. Find Image and put the answer in front of Image as a coefficient.

Image

Step 5. Review the main results.

Image

Image Exercise 3

For 1–4, find the two square roots of the given number.

1. 144

2. Image

3. 0.64

4. 400

For 5–18, find the indicated root, if possible.

5. Image

6. Image

7. Image

8. Image

9. Image

10. Image

11. Image

12. Image

13. Image

14. Image

15. Image

16. Image

17. Image

18. Image

For 19 and 20, simplify.

19. Image

20. Image