Core Focus onCore Focus onGeometryGeometry

Exponents and Roots

Lesson 4.5

Warm-UpSolve each equation.

1.

2.

3.

x = 9

x = −4

x = 100

5 3( 6) 11x

16 610x

2 7 25x

Exponents and RootsExponents and Roots

Lesson 4.5

Use roots to solve equations with exponents.

VocabularySquare RootOne of the two equal factors of a number.Symbol is .

Perfect SquareA number whose square root is an integer.

Cube RootOne of the three equal factors of a number.Symbol is .

Perfect CubeA number whose cube root is an integer.

3

The First 10Perfect Cubes

When finding the root of a number, the answer may be positive, negative or both. For example, if x2 = 9, then x can either equal 3 or –3 because (3)(3) = 9 and (−3)(−3) = 9.

When solving problems involving square and cube roots, consider the following rules:

Terms with exponents that are even will have two root answers, positive and negative. This is shown with the symbol ±.

Terms with exponents that are odd will have one root answer. It will have the same sign as the term under the root.

In real-world situations, negative answers may not always make sense. Look closely at the situation to determine if only a positive answer is needed.

Good to Know!

Using Roots to Solve Equations with Exponents

1. Isolate the variable with the exponent.

2. Use inverse operations to undo the exponent with the corresponding root.

3. Determine if the answer should be positive, negative or both based on the original exponent and the application setting.

Example 1Solve for x. Include all answers.

Subtract 11 from both sides of the equation.

Multiply both sides of the equation by 2.

Square root both sides of the equation.

Include both positive and negative answers.

2

11 432

x

2

11 432

x

11112

2 32 22

x

2 64x2 64x

8x

(–8)(–8) = 64and

(8)(8) = 64

Example 2Solve 2x3 − 5 = −59 for x.

Add 5 to both sides of the equation.

Divide both sides of the equation by 2.

Cube root both sides of the equation.

32 5 59 x5 532 54x

2 2 3 27x3 3 3 27 x

3x

(–3)(–3)(–3) = –27

Example 3The volume of a sphere can be found using the formula .Finley has a spherical balloon filled with water. She knowsthere are 56 cubic inches of water in the balloon. What is theapproximate radius, r, of the balloon?

Use 3.14 for π and round the answer to the nearest hundredth.

Write the formula for the volume of a sphere.

Substitute all known values for the variables.

Multiply both sides of the equation by thereciprocal of .

3

3

3

3

4343

3 4 34 3 4

56 (3.14)

56 (3.14)

42 3.14

V r

r

r

r43

343

V r

Example 3 Continued…The volume of a sphere can be found using the formula .Finley has a spherical balloon filled with water. She knowsthere are 56 cubic inches of water in the balloon. What is theapproximate radius, r, of the balloon?

Use 3.14 for π and round the answer to the nearest hundredth.

Divide both sides of the equation by 3.14.

Cube root both sides of the equation.

The radius of the balloon is approximately 2.37 inches.

342 3.14r3.14 3.14

313.38 r33 313.38 r

2.37 r

343

V r

Some equations in this lesson have one solution and others have two solutions. Is it possible to tell how many solutions an equation will have prior to solving? If so, explain how.

Communication Prompt

Exit ProblemsSolve each equation. Include all answers.

1. 2.

3. 4.

28 128x3 343x

2 2 225

x 34 72 572 x

x = 7 x = ±4

x = ±10 x = −5