Simplifying Square Roots Section 10.1 Simplifying Square Roots

Goal Simplify square roots.

Key Vocabulary Radical Radicand

Square Roots Opposite of squaring a number is taking the square root of a number. A number b is a square root of a number a if b2 = a. In order to find a square root of a, you need a # that, when squared, equals a.

22 = 4 2 2

2 The square root of 4 is

32 = 9 3 3

3 The square root of 9 is

42 = 16 4 4

4 The square root of 16 is

52 = 25 5 5

5 The square root of 25 is

Principal Square Roots Any positive number has two real square roots, one positive and one negative, √x and -√x √4 = 2 and -2, since 22 = 4 and (-2)2 = 4 The principal (positive) square root is noted as The negative square root is noted as

Radicands Radical expression is an expression containing a radical sign. Radicand is the expression under a radical sign. Note that if the radicand of a square root is a negative number, the radical is NOT a real number.

Perfect Squares Square roots of perfect square radicands simplify to rational numbers (numbers that can be written as a quotient of integers). Square roots of numbers that are not perfect squares (like 7, 10, etc.) are irrational numbers. IF REQUESTED, you can find a decimal approximation for these irrational numbers. Otherwise, leave them in radical form.

Calculator keystrokes Example 1 Use a Calculator to Find Square Roots Find the square root of 52. Round your answer to the nearest tenth. Check that your answer is reasonable. SOLUTION Calculator keystrokes 52 or 52 Display 7.21110 Rounded value ≈ 7.2 52 This is reasonable, because 52 is between the perfect squares 49 and 64. So, should be between and , or 7 and 8. The answer 7.2 is between 7 and 8. 52 64 49 16

These numbers are called the Perfect Squares. The terms of the following sequence: 1, 4, 9, 16, 25, 36, 49, 64, 81… 12,22,32,42, 52 , 62 , 72 , 82 , 92… These numbers are called the Perfect Squares.

Properties of Square Roots Properties of Square Roots (a, b > 0) Product Property Quotient Property

Product Property Example Use the product property to simplify the following radical expressions. No perfect square factor, so the radical is already simplified.

Quotient Property Example Use the quotient property to simplify the following radical expression.

Sums and Differences The product and quotient properties allow us to split radicals that have a radicand which is a product or a quotient. We can NOT split sums or differences.

Like Radicals “Like Terms” are terms with the same variables raised to the same powers. They can be combined through addition and subtraction. Similarly, we can work with the concept of “like” radicals to combine radicals with the same radicand. Like radicals are radicals with the same index and the same radicand. Like radicals can also be combined with addition or subtraction by using the distributive property.

Adding and Subtracting Radical Expressions Can not simplify Can not simplify

Adding and Subtracting Radical Expressions Simplify the following radical expression.

Multiplying Radicals To multiply radicals …multiply the inside by the inside and the outside by the outside. Then simplify.

Multiplying Radicals It may be easier to simplify the radicals first. 16 5 16 2 9 2 9 3 4 2 4 5

Multiply the radicals. Then simplify if possible. Example 2 Multiply Radicals Multiply the radicals. Then simplify if possible. b. · 2 8 a. 3 7 SOLUTION a. · 3 7 = b. = · 2 8 = 21 = 16 = 4 27

Squaring a Radical Remember “squaring” and “square root” are inverse operations.

Squaring a Radical Evaluate the expression Evaluate the expression

Example 3 Find Side Lengths Use the Pythagorean Theorem to find the length of the hypotenuse to the nearest tenth. SOLUTION a2 + b2 = c2 Write Pythagorean Theorem. + = c2 2 3 Substitute for a and for b. 2 + 3 = c2 Simplify. 5 = c2 Add. Take the square root of each side. = c 5 Use a calculator. 2.2 ≈ c 30

Simplifying Division of Radicals When dividing rational expressions with radical components, and the denominator is a factor of the numerator, consider the radicands and the coefficients separately. Simplify the radicands and reduce the coefficients. 2 4 11

Simplifying Square Root The properties of square roots allow us to simplify radical expressions. A radical expression is in simplest form when: The radicand has no perfect-square factor other than 1 There’s no radical in the denominator (rationalizing the denominator)

Simplifying Radical Expressions Step #1: The radicand has no perfect-square factor other than 1. Simplifying Radical Expressions

Simplest Radical Form Method 1: Factoring out a perfect square. Like the number 3/6, is not in its simplest form. Also, the process of simplification for both numbers involves factors.

Example Method 1 Simplifying Radicals 1. Find the largest perfect square that is a factor of the radicand. 2. Rewrite the radicand as a product of its largest square and some other number. 3. Take the square root of the perfect square. Write it as a product. 4. Leave the number that you didn’t take the square root of under the radical sign. Perfect squares: 1, 4 , 9, 16, 25, 36, 49, 64, 81, 100,...

Simplest Radical Form Method 2: Making a factor tree. In the second method, pairs of factors come out of the radical as single factors, but single factors stay within the radical.

Simplest Radical Form Method 2: Making a factor tree. This method works because pairs of factors are really perfect squares. So 5·5 is 52, the square root of which is 5.

Examples: Simplifying Radicals 25 2 144 2 11 11 25 6

Simplify the radical expression. Example 4 Simplify Radicals Simplify the radical expression. a. 12 b. 45 SOLUTION = a. 12 · 4 3 b. 45 · 9 5 = = · 4 3 = · 9 5 = 2 3 = 3 5 39

Examples: Simplifying Radicals 9 3 100 2 8 8 16 5

Your Turn: Express each square root in its simplest form by factoring out a perfect square or by using a factor tree.

Your Turn: Simplify the expression.

Simplifying Radical Expressions Step #2: There’s no radical in the denominator (rationalizing the denominator). Simplifying Radical Expressions

Example 1 Evaluate, and then classify the product. (√5)(√5) = (2 + √5)(2 – √5) =

Conjugates The radical expressions a + √b and a – √b are called conjugates. The product of two conjugates is always a rational number

Example 2 Identify the conjugate of each of the following radical expressions: √7 5 – √11 √13 + 9

Rationalizing the Denominator Recall that a radical expression is not in simplest form if it has a radical in the denominator. How could we use conjugates to get rid of any radical in the denominator and why?

Rationalizing the Denominator We can use conjugates to get rid of radicals in the denominator: The process of multiplying the top and bottom of a radical expression by the conjugate of the denominator is called rationalizing the denominator. Fancy One Fancy One

Example 3 Simplify the expression.

Your Turn: Simplify the expression.

Practice

Your Turn: Find the square root. Round your answer to the nearest tenth. Check that your answer is reasonable. 1. 27 ANSWER 5.2; < < , so 5 < 5.2 < 6. 25 27 36 2. 46 ANSWER 6.8; < < , so 6 < 6.8 < 7. 46 36 49 3. 8 ANSWER 2.8; < < , so 2 < 2.8 < 3. 4 8 9 4. 97 ANSWER 9.8; < < , so 9 < 9.8 < 10. 97 81 100

Your Turn: Multiply the radicals. Then simplify if possible. 5. · 3 5 ANSWER 15 6. · 11 6 ANSWER 66 7. · 27 3 ANSWER 9 8. 5 · 3 15 ANSWER

Your Turn: Simplify the radical expression. 9. 20 ANSWER 2 5 10. 8 ANSWER 2 11. 75 ANSWER 5 3 12. 112 ANSWER 4 7

Assignment Pg. 539 – 541; #1 – 61odd