Evaluate nth Roots and Use Rational Exponents Lesson 3.1 Honors Algebra 2 Evaluate nth Roots and Use Rational Exponents Lesson 3.1

Goals Goal Rubric Evaluate nth roots Rational exponents Evaluate expressions with rational exponents Solve equations with nth roots Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary nth root of a Index of a radical

nth Roots You are probably familiar with finding the square root of a number. Square and square root, these two operations are inverses of each other. Similarly, there are roots that correspond to larger powers. 5 and –5 are square roots of 25 because 52 = 25 and (–5)2 = 25 2 is the cube root of 8 because 23 = 8. 2 and –2 are fourth roots of 16 because 24 = 16 and (–2)4 = 16. a is the nth root of b if an = b.

nth Roots A radical sign is used to indicate a root. The number under the radical sign is the radicand. The index gives the degree of the root.

nth Roots The nth root of a real number a can be written as the radical expression 𝑛 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.

nth Roots When a radical sign shows no index, it represents a square root. Reading Math

Find the indicated real nth root(s) of a. EXAMPLE 1 Find nth roots Find the indicated real nth root(s) of a. a. n = 3, a = –216 b. n = 4, a = 81 SOLUTION a. Because n = 3 is odd and a = –216 < 0, –216 has one real cube root. Because (–6)3 = –216, you can write = 3√ –216 = –6 or (–216)1/3 = –6. b. Because n = 4 is even and a = 81 > 0, 81 has two real fourth roots. Because 34 = 81 and (–3)4 = 81, you can write ±4√ 81 = ±3 or ±811/4 = ±3.

Find the indicated real nth root(s) of a. 1. n = 4, a = 625 Your Turn: for Example 1, Find the indicated real nth root(s) of a. 1. n = 4, a = 625 4√ 625 = ±5 or ±(625)1/4 = ±5 ANSWER 2. n = 6, a = 64 6√ 64 = ±2 or ±(64)1/6 = ±2 ANSWER

Find the indicated real nth root(s) of a. 3. n = 3, a = –64 Your Turn: for Example 1 Find the indicated real nth root(s) of a. 3. n = 3, a = –64 3√ –64 = –4 or (–64)1/3 = –4 ANSWER 4. n = 5, a = 243 5√ 243 = 3 or (243)1/5 = 3 ANSWER

A rational exponent is an exponent that can be expressed as 𝑚 𝑛 , where m and n are integers and n ≠ 0. Radical expressions can be written by using rational exponents. Rational Exponents

Rational Exponents Writing Math The denominator of a rational exponent becomes the index of the radical. Writing Math

Rational Exponents Rational exponents have the same properties as integer exponent.

Evaluate (a) 163/2 and (b) 32–3/5. EXAMPLE 2 Evaluate expressions with rational exponents Evaluate (a) 163/2 and (b) 32–3/5. SOLUTION Rational Exponent Form Radical Form a. 163/2 (161/2)3 = 43 = 64 = 163/2 16 √ ( )3 = 43 = 64 = = 1 323/5 = 1 (321/5)3 1 323/5 = 1 ( )3 5 √ = 32 b. 32–3/5 32–3/5 = 1 23 1 8 = = 1 23 1 8 =

Evaluate the expressions without using a calculator. 5. 45/2 Your Turn: for Example 2 Evaluate the expressions without using a calculator. 5. 45/2 32 ANSWER 6. 9–1/2 1 3 ANSWER

Evaluate the expressions without using a calculator. 7. 813/4 Your Turn: for Example 2 Evaluate the expressions without using a calculator. 7. 813/4 27 ANSWER 8. 17/8 1 ANSWER

Approximating Roots with a Calculator Write the root as a rational exponent. Use a graphing calculator to approximate: 4 5 3 First rewrite 4 5 3 as 5 3 4 . Then enter the following:

EXAMPLE 3 Approximate roots with a calculator Expression Keystrokes Display a. 91/5 9 1 5 1.551845574 b. 123/8 12 3 8 2.539176951 ( )3 4 √ c. = 73/4 7 7 3 4 4.303517071

Your Turn: for Example 3 Evaluate the expression using a calculator. Round the result to two decimal places when appropriate. 9. 42/5 1.74 ANSWER 10. 64 2/3 – 0.06 ANSWER 11. (4√ 16)5 32 ANSWER 12. (3√ –30)2 9.65 ANSWER

Solve Equations Using nth Root To solve simple equations involving xn; isolate the power, xn or (x – a)n term. take the nth root of each side.

Solve the equation. a. 4x5 = 128 x5 32 = x = √32 x 2 = EXAMPLE 4 Solve equations using nth roots Solve the equation. a. 4x5 = 128 x5 32 = Divide each side by 4. x = 5 √32 Take fifth root of each side. x 2 = Simplify.

EXAMPLE 4 b. (x – 3)4 = 21 x – 3 21 4 – = x – 21 + 3 √ x 21 + 3 = or – Solve equations using nth roots b. (x – 3)4 = 21 x – 3 √ 21 4 + – = Take fourth roots of each side. x 4 + – 21 + 3 = √ Add 3 to each side. x 4 21 + 3 = or – √ Write solutions separately. x 5.14 or 0.86 Use a calculator.

Your Turn: for Example 4 Solve the equation. Round the result to two decimal places when appropriate. 13. x3 = 64 x = 4 ANSWER 14. 1 2 x5 = 512 x = 4 ANSWER

Your Turn: for Example 4 Solve the equation. Round the result to two decimal places when appropriate. 15. 3x2 = 108 x = + 6 – ANSWER 16. 1 4 x3 = 2 x = 2 ANSWER

Your Turn: for Example 4 Solve the equation. Round the result to two decimal places when appropriate. 17. ( x – 2 )3 = –14 x = – 0.41 ANSWER 18. ( x + 5 )4 = 16 ANSWER x = – 3 or = –7

EXAMPLE 5 Use nth roots in problem solving Biology A study determined that the weight w (in grams) of coral cod near Palawan Island, Philippines, can be approximated using the model w = 0.0167ℓ 3 where l is the coral cod’s length (in centimeters). Estimate the length of a coral cod that weighs 200 grams.

A coral cod that weighs 200 grams is about 23 centimeters long. EXAMPLE 5 Use nth roots in problem solving SOLUTION w 0.0167 l 3 = Write model for weight. 200 0.0167 l 3 = Substitute 200 for w. 11,976 l 3 Divide each side by 0.0167. 11,976 3 √ l Take cube root of each side. 22.9 l Use a calculator. ANSWER A coral cod that weighs 200 grams is about 23 centimeters long.

Your Turn: for Example 5 WHAT IF? Use the information from Example 5 to estimate the length of a coral cod that has the given weight. 19. a. 275 grams about 25 cm ANSWER b. 340 grams about 27 cm ANSWER c. 450 grams about 30 cm ANSWER