Rational Exponents Test Answer Key with Scoring – Algebra 1

Question 1 Multiply monomials (2 points)

6 × (−2) = −12. a: 4 + 3 = 7. b: 1 + 5 = 6.

−12a⁷b⁶

Scoring: 1 point for the coefficient, 1 point for the exponents.

Question 2 Divide monomials (2 points)

20 ÷ 5 = 4; m: 9 − 4 = 5; n: 3 − 8 = −5.

4m⁵n⁻⁵ = 4m⁵ ÷ n⁵.

4m⁵n⁵

Scoring: 1 point for the exponents, 1 point for positive exponents.

Question 3 Power of a product (2 points)

(−2)⁴ = 16, (x³)⁴ = x¹², y⁴.

16x¹²y⁴

Scoring: 1 point for the coefficient 16, 1 point for the variables.

Question 4 Evaluate rational exponents (2 points)

(a) 27^2/3 = (∛27)² = 3² = 9.

(b) 4^−3/2 = 1 ÷ 4^3/2 = 1 ÷ (√4)³ = 1 ÷ 8.

(a) 9 (b) 1/8

Scoring: 1 point per part.

Question 5 Radical and exponent forms (2 points)

(a) ∛(x⁷) = x^7/3.

(b) a^5/2 = √(a⁵), or equivalently (√a)⁵.

(a) x^7/3 (b) √(a⁵)

Scoring: 1 point per part.

Question 6 Simplify square roots (3 points)

(a) √98 = √(49 · 2) = 7√2.

(b) √45 = √(9 · 5) = 3√5.

(c) √(20x⁶) = √(4 · 5 · (x³)²) = 2x³√5.

(a) 7√2 (b) 3√5 (c) 2x³√5

Scoring: 1 point per part.

Question 7 Rectangle with radicals (3 points)

(a) √12 = 2√3 and √27 = 3√3.

(b) Area = 2√3 · 3√3 = 6 · 3 = 18 cm².

(c) Perimeter = 2(2√3 + 3√3) = 10√3 cm.

18 cm²; 10√3 cm

Scoring: 1 point per part.

Question 8 Radioactive decay (3 points)

(a) (1/2)³ = 1/8, so A = 10 g.

(b) (1/2)^1/2 = 1/√2, so A = 80/√2 = 40√2 ≈ 56.57 g.

(c) The mass halves every 4 hours: 4 is the half-life.

10 g; 40√2 ≈ 56.57 g; half-life 4 hours

Scoring: 1 point per part.

Question 9 Locate a root (1 point)

7² = 49 and 8² = 64, and 49 < 55 < 64.

Between 7 and 8

Scoring: 1 point.