Standard form (a x 10^k) practice questions.

Write the number 0.00047 in the form a × 10^k, where 1 ≤ a < 10 and k ∈ ℤ.

• 4.7 × 10⁻³
• 47 × 10⁻⁵
• 4.7 × 10⁻⁴
• 4.7 × 10⁴

Answer: 4.7 × 10⁻⁴

The decimal point moves 4 places to the right to give 4.7, so k = −4. Note that 47 × 10⁻⁵ has the correct value but a = 47 is not in the required range 1 ≤ a < 10.

Calculate (3 × 10⁵) × (4 × 10⁻²), giving your answer in the form a × 10^k, where 1 ≤ a < 10 and k ∈ ℤ.

• 1.2 × 10⁴
• 1.2 × 10³
• 7 × 10³
• 1.2 × 10⁻¹⁰

Answer: 1.2 × 10⁴

Multiply coefficients and add exponents: 3 × 4 = 12 and 10⁵⁺⁽⁻²⁾ = 10³, so 12 × 10³ = 1.2 × 10⁴ after adjusting the coefficient into the range 1 ≤ a < 10.

Calculate (8 × 10⁶) ÷ (2 × 10⁻³), giving your answer in the form a × 10^k, where 1 ≤ a < 10 and k ∈ ℤ.

• 4 × 10³
• 4 × 10⁹
• 4 × 10⁻²
• 1.6 × 10¹⁰

Answer: 4 × 10⁹

Divide coefficients and subtract exponents: 8 ÷ 2 = 4 and 6 − (−3) = 9, so the answer is 4 × 10⁹. A common error is computing 6 − 3 = 3 by forgetting the negative sign.

Determine which of the following numbers is the largest.

• 9.9 × 10⁴
• 8.7 × 10⁴
• 9.99 × 10⁴
• 1.01 × 10⁵

Answer: 1.01 × 10⁵

Compare exponents first: 1.01 × 10⁵ = 101 000, which exceeds every option with exponent 4 (the largest of those is 9.99 × 10⁴ = 99 900). A larger exponent outweighs a larger coefficient.

Calculate (2.5 × 10³)², giving your answer in the form a × 10^k, where 1 ≤ a < 10 and k ∈ ℤ.

• 5 × 10⁶
• 6.25 × 10⁶
• 6.25 × 10⁹
• 2.5 × 10⁶

Answer: 6.25 × 10⁶

Square the coefficient and double the exponent: 2.5² = 6.25 and (10³)² = 10⁶. A frequent error is squaring the exponent (10⁹) instead of multiplying it by 2.

Calculate (4.2 × 10⁵) + (3 × 10⁴), giving your answer in the form a × 10^k, where 1 ≤ a < 10 and k ∈ ℤ.

• 4.5 × 10⁵
• 7.2 × 10⁵
• 7.2 × 10⁹
• 4.5 × 10⁴

Answer: 4.5 × 10⁵

To add, rewrite with the same exponent: 4.2 × 10⁵ + 0.3 × 10⁵ = 4.5 × 10⁵. Adding the coefficients directly (4.2 + 3 = 7.2) ignores the different powers of 10.