Rational Exponents Worksheet Answer Key – Algebra 1

Problem 1 Product of powers

Same base, so add the exponents: 3 + 5 = 8.

x⁸

Problem 2 Quotient of powers

Subtract the exponents: 9 − 4 = 5.

x⁵

Problem 3 Power of a power

Multiply the exponents: 2 × 6 = 12.

x¹²

Problem 4 Negative exponent

A negative exponent means a reciprocal: 3⁻² = 1 ÷ 3² = 1 ÷ 9.

19

Problem 5 Between two integers

4² = 16 and 5² = 25, and 16 < 20 < 25, so √20 lies between 4 and 5. A calculator gives about 4.47.

Between 4 and 5

Problem 6 Multiply monomials

Multiply the coefficients: 3 × 4 = 12. Add exponents of x: 2 + 5 = 7. Add exponents of y: 3 + 1 = 4.

12x⁷y⁴

Problem 7 Divide monomials

18 ÷ 6 = 3. a: 7 − 3 = 4. b: 2 − 5 = −3.

So 3a⁴b⁻³ = 3a⁴ ÷ b³.

3a⁴b³

Problem 8 Power of a product

(2x³)⁴ = 2⁴ · x¹² = 16x¹².

(−3x²)³ = (−3)³ · x⁶ = −27x⁶.

16x¹² and −27x⁶

Problem 9 Positive exponents only

x⁻³ moves to the denominator as x³, and z⁻⁴ moves to the numerator as z⁴.

y²z⁴x³

Problem 10 Rational exponent evaluation

8^2/3 = (∛8)² = 2² = 4. 16^3/4 = (⁴√16)³ = 2³ = 8.

4 and 8

Problem 11 Radical and exponent forms

(a) ∛(x⁵) = x^5/3. (b) √x = x^1/2. (c) y^3/2 = √(y³) = (√y)³.

(a) x^5/3 (b) x^1/2 (c) √(y³)

Problem 12 Simplify a square root

The largest perfect square factor of 72 is 36: √72 = √36 · √2 = 6√2.

Check: 6(1.4142) ≈ 8.485 and √72 ≈ 8.485.

6√2

Problem 13 Square side length

Side = √48 = √(16 · 3) = 4√3.

4 × 1.732 ≈ 6.93.

4√3 ≈ 6.93 inches

Problem 14 Radical with a variable

√(18x⁶) = √(9 · 2 · (x³)²) = 3 · x³ · √2.

3x³√2

Problem 15 Rectangle with radical sides

√8 = 2√2 and √18 = 3√2.

Area = √8 · √18 = √144 = 12 ft².

Perimeter = 2(2√2 + 3√2) = 10√2 ≈ 14.14 ft.

Area 12 ft²; perimeter 10√2 ft

Problem 16 File sizes

Total: 2¹³ · 2⁷ = 2²⁰ = 1,048,576 bytes.

Per drive: 2²⁰ ÷ 2⁵ = 2¹⁵ = 32,768 bytes.

2²⁰ = 1,048,576 bytes; 2¹⁵ = 32,768 bytes per drive

Problem 17 Square garden fence

Side = √200 = √(100 · 2) = 10√2 ≈ 14.14 ft.

Fence = 4 · 10√2 = 40√2 ≈ 56.57 ft.

Side 10√2 ft; fence 40√2 ≈ 56.57 ft

Problem 18 Half-life

t = 18: (1/2)³ = 1/8, so A = 64/8 = 8 g.

t = 9: (1/2)^9/6 = (1/2)^3/2 = 1/(2√2) = √2/4, so A = 64 · √2/4 = 16√2 ≈ 22.63 g.

8 g and 16√2 ≈ 22.63 g

Problem 19 Falling object

d = 72: t = √(72/16) = √72 ÷ 4 = 6√2 ÷ 4 = 3√2/2 ≈ 2.12 s.

d = 100: t = √(100/16) = 10/4 = 2.5 s.

3√2/2 ≈ 2.12 s; 2.5 s

Problem 20 Ladder against a wall

c² = 6² + 12² = 36 + 144 = 180, so c = √180 = √(36 · 5) = 6√5.

6√5 ≈ 13.42 feet