1-3 Square Roots Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

Warm Up Round to the nearest tenth. 1. 3.14 2. 1.97 1. 3.14 2. 1.97 Find each square root. 3. 4. Write each fraction in simplest form. 5. 6. Simplify. 7. 8. 3.1 2.0 4 25

Objectives Estimate square roots. Simplify, add, subtract, multiply, and divide square roots.

Vocabulary radical symbol radicand principal root rationalize the denominator like radical terms

The side length of a square is the square root of its area The side length of a square is the square root of its area. This relationship is shown by a radical symbol . 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.

Example 1: Estimating Square Roots Estimate to the nearest tenth. Find the two perfect squares that 27 lies between. < 5 < < 6 Find the two integers that lies between . Because 27 is closer to 25 than to 36, is close to 5 than to 6. Try 5.2: 5.22 = 27.04 Too high, try 5.1. 5.12 = 26.01 Too low Because 27 is closer to 27.04 than 26.01, is closer to 5.2 than to 5.1. Check On a calculator ≈ 5.1961524 ≈ 5.1 rounded to the nearest tenth. 

Estimate to the nearest tenth. Check It Out! Example 1 Estimate to the nearest tenth. Find the two perfect squares that –55 lies between. < –7 < < –8 Find the two integers that lies between – . Because –55 is closer to –49 than to –64, is closer to –7 than to –8. Try 7.2: 7.22 = 51.84 Too low, try 7.4 7.42 = 54.76 Too low but very close Because 55 is closer to 54.76 than 51.84, is closer to 7.4 than to 7.2. Check On 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 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 Simplify each expression. C. Product Property of Square Roots D. Quotient Property of Square Roots

Simplify each expression. Check It Out! 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.

Simplify each expression. Check It Out! 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 Simplify by rationalizing the denominator. Multiply by a form of 1. = 2

Example 3B: Rationalizing the Denominator Simplify the expression. Multiply by a form of 1.

Check It Out! Example 3a Simplify by rationalizing the denominator. Multiply by a form of 1.

Check It Out! Example 3b Simplify by rationalizing the denominator. Multiply by a form of 1.

Square roots that have the same radicand are called 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

Example 4B: Adding and Subtracting Square Roots Simplify radical terms. Combine like radical terms.

Check It Out! Example 4a Add or subtract. Combine like radical terms.

Check It Out! Example 4b Add or subtract. Simplify radical terms. Combine like radical terms.

6.7 Lesson Quiz: Part I 1. Estimate to the nearest tenth. Simplify each expression. 2. 3. 4. 5.

Lesson Quiz: Part II Simplify by rationalizing each denominator. 6. 7. Add or subtract. 8. 9.