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Foundations · Numbers

Powers, roots and scientific notation

Squares, cubes and powers, the index laws, square roots, and writing very large or very small numbers in scientific notation.

First, the rules

Repeated multiplication, and its shortcuts

2⁵ is five 2s multiplied. Every index law is just counting those 2s — which is why the laws work for letters as well as numbers.

★ Meaning
Example 1
an = a × a × … × a (n factors) · a¹ = a, a⁰ = 1
★ Multiply / divide same base
Example 1
am · an = am+n · am ÷ an = am−n
★ Power of a power / product
Example 1
(am)n = amn · (ab)n = anbn
★ Negative exponent
Example 2
a−n = 1an — 2⁻³ = 1/8; a minus in the exponent never makes the value negative.
★ Roots
Example 2
√a = the number whose square is a · ∛a: cube
√144 = 12, ∛27 = 3. Know the squares to 15² and cubes to 5³.
Simplifying roots
Example 2
√50 = √25 · √2 = 5√2 — pull out the largest square factor.
★ Scientific notation
Example 3
a × 10n, 1 ≤ a < 10
4 300 000 = 4.3 × 10⁶ · 0.00052 = 5.2 × 10⁻⁴. The exponent counts how many places the point moved.

See it

Watch a power grow

Each bar is the one before multiplied by the base. Change the base and the exponent and read the value, its scientific notation and its negative-exponent twin.

Step by step

Worked examples

1

Index laws

laws

Simplify (a) 3⁴ × 3⁵, (b) x⁸ ÷ x³, (c) (2y²)³.

Ask yourself: Count the factors in each case.

Show the solution
  1. (a) Nine 3s in total: 3⁹ (= 19 683).
  2. (b) Eight x's over three x's leaves five: x⁵.
  3. (c) Cube both parts: 2³ · (y²)³ = 8y⁶.

Answer 3⁹ · x⁵ · 8y⁶

2

Negatives and roots

negative exponent · roots

Evaluate (a) 5⁻², (b) √196 + ∛125, and simplify (c) √72.

Ask yourself: A negative exponent flips; for roots, look for squares and cubes.

Show the solution
  1. (a) 15² = 125 = 0.04.
  2. (b) 14² = 196 and 5³ = 125: 14 + 5 = 19.
  3. (c) 72 = 36 × 2: √72 = 6√2 ≈ 8.49.

Answer 125 · 19 · 6√2

3

Scientific notation

large and small

(a) Write 0.000 081 and 7 020 000 in scientific notation. (b) Work out (3 × 10⁴) × (2 × 10⁵).

Ask yourself: Move the point until one non-zero digit is in front; count the moves.

Show the solution
  1. (a) 8.1 with the point moved 5 places right: 8.1 × 10⁻⁵. 7.02 with the point moved 6 places left: 7.02 × 10⁶.
  2. (b) Multiply the numbers and add the exponents: 6 × 10⁹ — already in the right form: 6 × 10⁹.

Answer 8.1 × 10⁻⁵ · 7.02 × 10⁶ · 6 × 10⁹

Practice: powers and roots

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Evaluating powers, index laws, negative exponents, square and cube roots, scientific notation. Your best score is kept in this browser only.

About this lesson

Powers, roots and scientific notation: squares, cubes and higher powers, the laws of indices for multiplying and dividing powers, negative and zero exponents, square and cube roots, and scientific notation for very large and very small numbers. A growth bar you can move, worked examples and practice questions with explanations. Everything on this page runs in your browser; nothing you type is sent or stored. Spotted an error? Tell us.