Beginning Algebra, 11th Edition

OBJECTIVE 5 Use the distance formula.Consider

FIGURE 3. The distance between the points and

is and the distance between the

points and is From the

Pythagorean theorem,

Taking the square root of each side gives the distance

formula.

d^2 = 1 x 2 - x 122 + 1 y 2 - y 122.

1 x 1 , y 12 1 x 2 , y 12 b=x 2 - x 1.

1 x 2 , y 12 a=y 2 - y 1 ,

1 x 2 , y 22

SECTION 8.1 Evaluating Roots 499

a x

b

d

y

0

(x 2 , y 2 )

(x 1 , y 1 )(x 2 , y 1 )

FIGURE 3

Distance Formula

The distance between the points and is

d 1 x 2 x 122  1 y 2 y 122.

1 x 1 , y 12 1 x 2 , y 22

Using the Distance Formula

Find the distance between and

Distance formula

= 226

= 252 + 12

= 212 - 1 - 3222 + 15 - 422

d= 21 x 2 - x 122 + 1 y 2 - y 122

1 - 3, 4 2 1 2, 5 2.

NOW TRY EXAMPLE 8
EXERCISE 8
Find the distance between
1 - 4, 3 2 and. 1 - 7, 1 2

The nth root of a, written ,is a number whose nth power equals a. That is,

2 nab means bna.

n

2 a

n

2 a

NOW TRY

OBJECTIVE 6 Find cube, fourth, and other roots.Finding the square root of

a number is the opposite (inverse) of squaring a number. In a similar way, there are in-

verses to finding the cube of a number or to finding the fourth or greater power of a

number. These inverses are, respectively, the cube root,written and the fourth

root,written Similar symbols are used for other roots.

4

2 a.

3

2 a,

Let

and 1 x 2 , y 22 = 1 2, 5 2.

Start with the x-value^1 x^1 , y^12 =^1 - 3, 4^2

and the y-value of

the samepoint.

In the number nis the indexor orderof the radical.

Radical symbol Index Radicand

Radical

We could write instead of but the simpler symbol is customary, since

the square root is the most commonly used root.

NOTE When working with cube roots or fourth roots, it is helpful to memorize the

first few perfect cubes and so on and the first few perfect

fourth powers 124 =16, 34 =81,and so on. See 2 Exercises 93 and 94.

123 =8, 3^3 =27, 2

22 a 2 a, 2 a

n

2 a

2

n

a,

NOW TRY ANSWER

8. 213