Quantitative: Exponents and Roots Flashcards

Simplify: 2³ × 2⁵

Answer: 2⁸

When multiplying powers with the same base, add the exponents: 2³ × 2⁵ = 2^(3+5) = 2⁸.

Simplify: (3²)⁴

Answer: 3⁸

Power of a power rule: (aᵐ)ⁿ = a^(m×n). (3²)⁴ = 3^(2×4) = 3⁸.

What is the value of 5⁰?

Answer: 1

Any nonzero number raised to the power 0 equals 1. 5⁰ = 1.

What is the value of 2⁻³?

Answer: 1/8

Negative exponent rule: a⁻ⁿ = 1/aⁿ. 2⁻³ = 1/2³ = 1/8.

Simplify: 6⁸ ÷ 6³

Answer: 6⁵

Division rule: aᵐ ÷ aⁿ = a^(m−n). 6⁸ ÷ 6³ = 6^(8−3) = 6⁵.

What is √144?

Answer: 12

12² = 144, so √144 = 12.

Simplify: √(50)

Answer: 5√2

√50 = √(25 × 2) = √25 × √2 = 5√2.

What is 27^(1/3)?

Answer: 3

27^(1/3) is the cube root of 27. Since 3³ = 27, the answer is 3.

Simplify: (2x³)(4x²)

Answer: 8x⁵

Multiply coefficients: 2 × 4 = 8. Multiply powers: x³ × x² = x^(3+2) = x⁵. Result: 8x⁵.

If 2^x = 32, what is x?

Answer: 5

32 = 2⁵, so x = 5.

Simplify: (x⁴)/(x⁷)

Answer: 1/x³

x⁴/x⁷ = x^(4−7) = x⁻³ = 1/x³.

What is the value of (1/2)⁻²?

Answer: 4

(1/2)⁻² = (2/1)² = 2² = 4. A negative exponent flips the fraction.

Which is greater: 2¹⁰ or 10²?

Answer: 2¹⁰

2¹⁰ = 1,024. 10² = 100. 1,024 > 100, so 2¹⁰ is greater.

Simplify: √(x⁶)

Answer: x³

√(x⁶) = x^(6/2) = x³ (assuming x ≥ 0).

If 3^(2x) = 81, what is x?

Answer: 2

81 = 3⁴. So 3^(2x) = 3⁴ → 2x = 4 → x = 2.

Simplify: (2³ × 3²) / (2 × 3)

Answer: 12

(2³/2) × (3²/3) = 2² × 3¹ = 4 × 3 = 12.

What is 4^(3/2)?

Answer: 8

4^(3/2) = (4^(1/2))³ = (√4)³ = 2³ = 8.

Simplify: (√3 + √3)²

Answer: 12

√3 + √3 = 2√3. (2√3)² = 4 × 3 = 12.

If x² = 49 and x > 0, what is x³?

Answer: 343

x² = 49 → x = 7 (since x > 0). x³ = 7³ = 343.

Which expression equals √(a²b⁴) where a, b > 0?

Answer: ab²

√(a²b⁴) = √(a²) × √(b⁴) = a × b² = ab².