Beginning Algebra, 11th Edition

Approximating Irrational Square Roots

Find a decimal approximation for each square root. Round answers to the nearest

thousandth.

(a)

Using the square root key on a calculator gives where the

symbol means “is approximately equal to.”

(b) Use a calculator. (c) NOW TRY

OBJECTIVE 4 Use the Pythagorean theorem.

Many applications of square roots require the use of the

Pythagorean theorem. Recall from Section 6.6 and

FIGURE 1that if cis the length of the hypotenuse of a right

triangle, and aand bare the lengths of the two legs, then

In the next example, we use the fact that if then the positive solution of

the equation is (See page 554.)

Using the Pythagorean Theorem

Find the length of the unknown side in each right triangle with sides a, b, and c,

where cis the hypotenuse.

(a)

Use the Pythagorean theorem.

Let and.

Square.

Add.

Since the length of a side of a triangle must be a positive number, find the positive

square root of 25 to get c.

(b)

Use the Pythagorean theorem.

Let and.

Square.

Subtract 25.

Use a calculator to approximate the positive square root of 56.

a= 256 L7.483 NOW TRY

a^2 = 56

a^2 + 25 = 81

a 2 + 52 = 92 b= 5 c= 9

a^2 +b^2 =c^2

b=5, c= 9

c= 225 = 5

25 =c^2

9 + 16 =c^2

3 2 + 42 =c^2 a= 3 b= 4

a^2 +b^2 =c^2

a=3, b= 4

EXAMPLE 6

x^2 =k 2 k.

k 7 0,

a^2 b^2 c^2.

239 L6.245 - 2740 L -27.203

«

3.31662479L3.317,

211

EXAMPLE 5

SECTION 8.1 Evaluating Roots 497

Solve for .a^2

CAUTION Be careful not to make the common mistake of thinking that

equals Consider the following.

but

In general, 2 a^2 +b^2 Za+b.

29 + 16 = 225 = 5 , 29 + 216 = 3 + 4 = 7.

2 a^2 +b^2 a+b.

NOW TRY
EXERCISE 5
Find a decimal approximation
for each square root. Round
answers to the nearest thou-
sandth.

(a) 251 (b) - 2360

Hypotenuse

Leg a c

90 

Leg b

FIGURE 1

NOW TRY
EXERCISE 6
Find the length of the
unknown side in each right
triangle with sides a, b,and c,
where cis the hypotenuse.
Give any decimal approxima-
tions to the nearest
thousandth.

(a)

(b)a=9, c= 14

a=5, b= 12

NOW TRY ANSWERS

5. (a)7.141 (b)


6. (a)c= 13 (b)bL10.724
   
   
   • 18.974