Beginning Algebra, 11th Edition

OBJECTIVE 2 Decide whether a given root is rational, irrational, or not a

real number.Numbers with square roots that are rational are called perfect squares.

Perfect squares Rational square roots

25

144 are perfect squares since

A number that is not a perfect square has a square root that is not a rational num-

ber. For example, is not a rational number because it cannot be written as the

ratio of two integers. Its decimal equivalent neither terminates nor repeats. However,

is a real number and corresponds to a point on the number line.

As mentioned in Section 1.4,a real number that is not rational is called an

irrational number.The number is irrational. Many square roots of integers are

irrational.

25

25

25

B

4

9

=

2

3

4

9

2144 = 12

225 = 5

496 CHAPTER 8 Roots and Radicals

If ais a positive real number that is nota perfect square, then 2 ais irrational.

If ais a negativereal number, then 2 ais nota real number.

CAUTION Do not confuse and is not a real number,

since there is no real number that can be squared to obtain. However, is

the negative square root of 36, which is - 6.

- 36 - 236

2 - 36 - 236. 2 - 36

Not every number has a real number square root.For example, there is no real

number that can be squared to get. (The square of a real number can never be

negative.) Because of this, 2  36 is not a real number.

- 36

Identifying Types of Square Roots

Tell whether each square root is rational, irrational,or not a real number.

(a) Because 17 is not a perfect square, is irrational.

(b) The number 64 is a perfect square, so a rational number.

(c) There is no real number whose square is. Therefore, is not

a real number.

2 - 25 - 25 2 - 25

264 82 , 264 =8,

217 217

NOW TRY EXAMPLE 4
EXERCISE 4
Tell whether each square root
is rational, irrational,or not a
real number.

(a) (b)

(c) 2 - 16

231 2900

NOW TRY

NOTE Not all irrational numbers are square roots of integers. For example, (ap-

proximately 3.14159) is an irrational number that is not a square root of any integer.

OBJECTIVE 3 Find decimal approximations for irrational square roots.

Even if a number is irrational, a decimal that approximatesthe number can be found

using a calculator.

p

NOW TRY ANSWERS

4. (a)irrational (b)rational
   (c)not a real number

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