Laws of exponents (integer); introduction to logarithms (base 10, e) practice questions.

Simplify (3x⁴)² × x⁻⁵, giving your answer in the form kxⁿ.

• 3x³
• 9x¹³
• 9x³
• 6x³

Answer: 9x³

(3x⁴)² = 9x⁸, and multiplying by x⁻⁵ means adding the exponents: 9x⁸⁻⁵ = 9x³. Remember to square the coefficient 3 as well.

Calculate the exact value of 5⁰ + 5⁻¹.

• 1.2
• 0.2
• 5.2
• 6

Answer: 1.2

Any non-zero number to the power 0 equals 1, and 5⁻¹ = 1/5 = 0.2, so the sum is 1 + 0.2 = 1.2.

Which of the following is equal to (2⁻³)²?

• 1/32
• 64
• −64
• 1/64

Answer: 1/64

Raising a power to a power multiplies the exponents: (2⁻³)² = 2⁻⁶ = 1/2⁶ = 1/64. A negative exponent gives a reciprocal, never a negative value.

Given that 10ᵏ = 0.001, determine the value of k.

• 3
• −3
• −4
• 0.3

Answer: −3

0.001 = 1/1000 = 10⁻³, so k = −3. Equivalently, k = log₁₀ 0.001 = −3.

The equation eˣ = 8 has exact solution:

• x = ln 8
• x = log₁₀ 8
• x = 8/e
• x = e⁸

Answer: x = ln 8

The natural logarithm ln is the inverse of the exponential with base e, so eˣ = 8 gives x = ln 8 (≈ 2.08). log₁₀ 8 would solve 10ˣ = 8 instead.

Simplify (4a³b⁻²) ÷ (2a⁻¹b).

• 2a²b⁻¹
• 2a³b⁻²
• 2a⁴b⁻³
• 8a⁴b⁻³

Answer: 2a⁴b⁻³

Divide the coefficients (4 ÷ 2 = 2) and subtract exponents of like bases: a³⁻⁽⁻¹⁾ = a⁴ and b⁻²⁻¹ = b⁻³, giving 2a⁴b⁻³.