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Round to the nearest tenth.

1. 3.14 2. 1.97

Find each square root.

3. 4.

Write each fraction in simplest form.

5. 6.

Simplify.

7. 8.

2.0

3.1

25

4

The side length of a square is the square root of its area. This relationship is shown by a radicalsymbol . The number or expression under the radical symbol is called the radicand. The radical symbol indicates only the positive square root of a number, called the principal root. To indicate both the positive and negative square roots of a number, use the plus or minus sign (±).

or –5

Numbers such as 25 that have integer square roots are called perfect squares. Square roots of integers that are not perfect squares are irrational numbers. You can estimate the value of these square roots by comparing them with perfect squares. For example, lies between and , so it lies between 2 and 3.

<

< 6

5 <

Because 27 is closer to 25 than to 36, is close to 5 than to 6.

Because 27 is closer to 27.04 than 26.01, is closer to 5.2 than to 5.1.

Example 1: Estimating Square Roots

Estimate to the nearest tenth.

Find the two perfect squares that 27 lies between.

Find the two integers that lies between .

Try 5.2: 5.22 = 27.04

Too high, try 5.1.

5.12 = 26.01

Too low

Check On a calculator ≈ 5.1961524 ≈ 5.1 rounded to the nearest tenth.

<

< –8

–7 <

Because –55 is closer to –49 than to –64, is closer to –7 than to –8.

Because 55 is closer to 54.76 than 51.84, is closer to 7.4 than to 7.2.

Check It Out! Example 1

Estimate to the nearest tenth.

Find the two perfect squares that –55 lies between.

Find the two integers that lies between – .

Try 7.2: 7.22 = 51.84

Too low, try 7.4

7.42 = 54.76

Too low but very close

CheckOn a calculator ≈ –7.4161984 ≈ –7.4 rounded to the nearest tenth.

Square roots have special properties that help you simplify, multiply, and divide them.

Notice that these properties can be used to combine quantities under the radical symbol or separate them for the purpose of simplifying square-root expressions. A square-root expression is in simplest form when the radicand has no perfect-square factors (except 1) and there are no radicals in the denominator.

Example 2: Simplifying Square–Root Expressions quantities under the radical symbol or separate them for the purpose of simplifying square-root expressions. A square-root expression is in simplest form when the radicand has no perfect-square factors (except 1) and there are no radicals in the denominator.

Simplify each expression.

A.

Find a perfect square factor of 32.

Product Property of Square Roots

B.

Quotient Property of Square Roots

Example 2: Simplifying Square–Root Expressions quantities under the radical symbol or separate them for the purpose of simplifying square-root expressions. A square-root expression is in simplest form when the radicand has no perfect-square factors (except 1) and there are no radicals in the denominator.

Simplify each expression.

C.

Product Property of Square Roots

D.

Quotient Property of Square Roots

Check It Out! quantities under the radical symbol or separate them for the purpose of simplifying square-root expressions. A square-root expression is in simplest form when the radicand has no perfect-square factors (except 1) and there are no radicals in the denominator.Example 2

Simplify each expression.

A.

Find a perfect square factor of 48.

Product Property of Square Roots

B.

Quotient Property of Square Roots

Simplify.

Check It Out! quantities under the radical symbol or separate them for the purpose of simplifying square-root expressions. A square-root expression is in simplest form when the radicand has no perfect-square factors (except 1) and there are no radicals in the denominator.Example 2

Simplify each expression.

C.

Product Property of Square Roots

D.

Quotient Property of Square Roots

If a fraction has a denominator that is a square root, you can simplify it by rationalizing the denominator. To do this, multiply both the numerator and denominator by a number that produces a perfect square under the radical sign in the denominator.

Example 3A: Rationalizing the Denominator can simplify it by

Simplify by rationalizing the denominator.

Multiply by a form of 1.

= 2

Example 3B: Rationalizing the Denominator can simplify it by

Simplify the expression.

Multiply by a form of 1.

Check It Out! can simplify it by Example 3a

Simplify by rationalizing the denominator.

Multiply by a form of 1.

Check It Out! can simplify it by Example 3b

Simplify by rationalizing the denominator.

Multiply by a form of 1.

Square roots that have the same radicand are called can simplify it by like radical terms.

To add or subtract square roots, first simplify each radical term and then combine like radical terms by adding or subtracting their coefficients.

Example 4A: Adding and Subtracting Square Roots can simplify it by

Add.

Example 4B: Adding and Subtracting Square Roots can simplify it by

Subtract.

Simplify radical terms.

Combine like radical terms.

Check It Out! can simplify it by Example 4b

Add or subtract.

Simplify radical terms.

Combine like radical terms.

Lesson Quiz: Part I can simplify it by

1. Estimate to the nearest tenth.

6.7

Simplify each expression.

2.

3.

4.

5.

Lesson Quiz: Part II can simplify it by

Simplify by rationalizing each denominator.

6.

7.

Add or subtract.

8.

9.

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