5-6 Study Guide. Radical Expressions and Rational Exponents. Attendance Problems. Simplify each expression. (No decimal answers!

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1 Page 1 of 12 Radical Expressions and Rational Exponents Attendance Problems. Simplify each expression. (No decimal answers) I can rewrite radical expressions by using rational exponents. I can simplify and evaluate radical expressions and expressions containing rational exponents index Vocabulary rational exponent ( ) 7 Common Core: CCSS.MATH.CONTENT.HSA.REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. You are probably familiar with finding the square root of a number. These two operations are inverses of each other. Similarly, there are roots that correspond to larger powers.

2 Page 2 of 12 is the square root of 2 because 2 = 2. 2 is the cube root of 8 because 2 = 8. 2 is the fourth root of 16 because 2 = 16. a is the nth root of b if a n = b. n The nth root of a real number a can be written as the radical expression a, where n is the index (plural: indices) of the radical and a is the radicand. When a number has more than one root, the radical sign indicates only the principal, or positive, root. Video Example 1. With a domain of the real numbers, A. Find all the square roots of 2. B. Find the all the fifth roots of 2. Reading Math When a radical sign shows no index, it represents a square root.

3 Page of 12 C. Find all the cube roots of -6. D. Find all the fourth roots of Finding Real Roots Find all real roots. A fourth roots of 81 A positive number has two real fourth roots. Because = 81 and (-) = 81, the roots are and -. B cube roots of -12 A negative number has one real cube root. Because (-) = -12, the root is -. C sixth roots of -729 A negative number has no real sixth roots. Example 1. With a domain of the real numbers, A. sixth roots of 6. B. cube roots of C. fourth roots of -102.

4 Page of 12 Guided Practice. With a domain of the real numbers,. Find all the fourth roots of Find all the sixth roots of Find all the cube roots of 12. Video Example 2. Simplify each expression. Assume that all variables are positive. Remember When an expression contains a radical in the denominator, you must rationalize the denominator. To do so, rewrite the expression so that the denominator contains no radicals. A. 16x 12 B. w

5 Page of 12 2 Simplifying Radical Expressions Simplify each expression. Assume that all variables are positive. A 27 x 6 B _ x 7 x x Factor into perfect cubes. _ x 7 Quotient Property x x Product Property x_ 7 x x Simplify. x_ x 2 x _ x _ 9 7 Simplify the numerator. Rationalize the denominator. Product Property Simplify. Example 2. Simplify each expression. Assume that all variables are positive. 81x 12 B. 16x 8 Guided Practice. Simplify each expression. Assume that all variables are positive x x 7 x 2 x 8

6 Page 6 of 12 m A rational exponent is an exponent that can be expressed as, where m and n n are integers and n 0. Radical expressions can be written by using rational exponents. Video Example. Write ( 8) in radical form and simplify. Writing Math The denominator of a rational exponent becomes the index of the radical. Writing Expressions in Radical Form 2_ Write the expression (-12) in radical form, and simplify. Method 1 Evaluate the root first. ( -12) 2 Write with a radical. Method 2 Evaluate the power first. (-12) 2 Write with a radical. (-) 2 Evaluate the root. 2 Evaluate the power. 1,62 Evaluate the power. 2 Evaluate the root.

7 Page 7 of 12 Example. Write the expression 2 in radical form and simplify. Guided Practice. Write each expression in radical form, and simplify Video Example. Write each expression by using rational exponents ( ) Writing Expressions by Using Rational Exponents Write each expression by using rational exponents. A 7 B 7 _ 11 6 n a m = a m_ n 11 6_ n a m = a m_ n 11 2 = 121 Simplify. Example. Write each expression by using rational exponents. A. 1 B. 8 1

8 Page 8 of 12 Guided Practice. Write each expression by using rational exponents. ( )

9 Page 9 of 12 Video Example. Simplify each expression. A B Simplifying Expressions with Rational Exponents Simplify each expression. _ 2_ A _ + _ 2 Product of Powers 2 1 Simplify. 2 Evaluate the power. Check Enter the expression in a graphing calculator. B 1 8 2_ 8 8 1_ - _ 2 Quotient of Powers 8 - _ 1 Simplify. 1_ Negative Exponent 1_ 8 Property 1_ Evaluate the power. 2 Check Enter the expression in a graphing calculator. Example. Simplify each expression A B. 16

10 Page 10 of 12 Guided Practice. Simplify each expression Video Example 6. Frets are small metal bars positioned across the neck of a guitar so that the guitar can produce the notes of a specific scale. To find the distance a 12 fret should be placed from the bridge, multiply the length of the string by, where n is the number of notes higher than the string s root note. Where should a fret be placed to produce an A flat note on the E string ( notes higher)? ( ) n

11 Page 11 of 12 6 Music Application Frets are small metal bars positioned across the neck of a guitar so that the guitar can produce the notes of a specific scale. To find the distance a fret should be placed from the bridge, multiply the length of the string by 2 - n_ 12, where n is the number of notes higher than the string s root note. Where should a fret be placed to produce a G note on the E string ( notes higher)? 6 ( 2 - n _ 12 ) = 6 ( 2 - _ 12 ) Use 6 cm for the length of the string, and substitute for n. = 6 ( 2 - _ 1 ) Simplify. = 6 ( 1 _ = 6 _ 2 1_ 2.82 ) 1_ Negative Exponent Property Simplify. Use a calculator. Bridge The fret should be placed about.82 cm from the bridge. 6 cm E string Frets Example 6. Radium-226 is a form of radioactive element that decays over time. An initial sample of radium-226 has a mass of 00 mg. The mass of radium-226 remaining from the initial sample after t years is given by 00 2 t To the nearest milligram, how much radium-226 would be left after 800 years?

12 Page 12 of Guided Practice. To find the distance a fret should be place from the bridge on a guitar, multiply the length of the string by, where n is the number of notes higher that the string s root note. Where should the fret be placed to produce the E note that is one octave higher on the E string (12 notes higher)? -6 Assignment (p 6) -7 odd, 6-71 odd. 2 n 12

Radical Expressions and Graph (7.1) EXAMPLE #1: EXAMPLE #2: EXAMPLE #3: Find roots of numbers (Objective #1) Figure #1:

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