Grade 9 Exponents and Scientific Notation

Key concepts

• aᵐ × aⁿ = aᵐ⁺ⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• (aᵐ)ⁿ = aᵐⁿ and (ab)ⁿ = aⁿbⁿ
• a⁰ = 1 and a⁻ⁿ = 1/aⁿ
• Square roots and cube roots of perfect squares and cubes
• Scientific notation: a × 10ⁿ with 1 ≤ a < 10

Worked examples

Simplify: (2x³)²

Square each factor: 2² = 4 and (x³)² = x⁶. So (2x³)² = 4x⁶.

Write 0.000 45 in scientific notation.

Move the decimal comma 4 places to the right to get 4.5. Because the number is small, the power is negative: 4.5 × 10⁻⁴.

Simplify: 3a⁵ × 2a⁻²

Multiply the numbers and add the exponents: 3 × 2 = 6 and a⁵⁺⁽⁻²⁾ = a³. So the answer is 6a³.

Calculate: (4 × 10³) × (3 × 10⁵)

Multiply the numbers and add the powers of 10: 12 × 10⁸. Write it in scientific notation: 1.2 × 10⁹.

Common mistakes

• Multiplying the exponents when you should add them (a² × a³ = a⁵, not a⁶).
• Thinking a negative exponent makes the answer negative: 2⁻³ = 1/8.
• Writing 12 × 10⁸ as the final answer; scientific notation needs 1 ≤ a < 10.
• Squaring only the variable: (2x)² is 4x², not 2x².

Exam tips

• Write each exponent law next to your step the first few times you use it.
• Count the places the decimal comma moves to get the power of 10.
• Check simple cases with numbers: if x = 2, does your simplified answer give the same value?

Frequently asked questions

Why is anything to the power of 0 equal to 1?

Because aⁿ ÷ aⁿ = 1, and by the division law it is also aⁿ⁻ⁿ = a⁰. So a⁰ = 1 for any a that isn't zero.

When do we use scientific notation?

For very big or very small numbers, such as the distance to the sun (1.5 × 10⁸ km) or the size of a cell. It makes them easier to read and compare.

What is the difference between a square root and a cube root?

The square root of 49 is 7 because 7 × 7 = 49. The cube root of 27 is 3 because 3 × 3 × 3 = 27.