Algebra 2 Chapter 5 Review

Key Concepts

• Focus on radical and exponential expressions.
• Roots and powers undo each other.
• Properties of radicals and exponents.

Solving Real Solutions

Problem 1

• Isolate the term with an exponent.
• Multiply both sides by 9: x^3 = 729.
• Take the cubed root of both sides: x = 9.

Problem 2

• Add 7 to both sides: (x - 11)^4 = 256.
• Take the fourth root (even root, remember ±): x - 11 = ±4.
• Solve for x: x = 7 or 15 by adding 11.

Properties of Exponents and Radicals

Problem 3

• Show work, no calculator: 32^(2/5).
• Break it into root and power: 2^5 = 32, thus (2^5)^(2/5) = 2^2 = 4.

Problem 4

• Simplify division under one root: (32/2)^(1/4) = 16^(1/4) = 2.

Problem 5

• Seek perfect cubes.
• Break 16 into factor tree: 2^4.
• Simplify: free a 2, cube root of c^5 results in c^(5/3).

Combination of Roots

Problem 6

• Combined under one root: Cube root of 136.

Problem 7

• Adding radicals: Combine like terms after simplifying.

Rationalizing the Denominator

Problem 8

• Conjugate multiplication to rationalize: 15 - 5√2 / 7.

Problem 9

• Use cubed root to rationalize: Cube root of x^3 = x.

Graphing and Transformations

Problem 12

• Square root graph: Shift left 3, down 2.
• Domain: x ≥ -3, Range: y ≥ -2.

Problem 13

• Cube root graph: Faster growth and shift.
• Domain and range are all real numbers.

Problem Translations and Transformations

Problem 14

• Horizontal stretch, vertical flips, shifts.

Problem 15

• Apply transformations step by step.

Solving Radical Equations

Problem 16-18

• Convert exponents to radicals.
• Isolate and solve step by step.
• Verify solutions, especially with even powers.

Inequalities and Domains

Problems 19-20

• Solve inequalities considering domain restrictions.
• Use a number line for visualization.

Combining Functions

Problem 21-22

• Add, subtract, and multiply functions.
• Use given values to test expressions.

Inverses of Functions

Problems 23-25

• Use f(g(x)) and g(f(x)) to verify inverses.
• Graph functions to check symmetry.