Powers and Roots
Powers and Roots
Index notation provides a compact way to express repeated multiplication. The expression \( a^n \) means \( a \) multiplied by itself \( n \) times: the base is \( a \) and the index (also called the exponent or power) is \( n \). For example, \( 5^3 = 5 \times 5 \times 5 = 125 \).
Roots are the inverse of powers. The square root \( \sqrt{a} \) is the positive number whose square equals \( a \); the cube root \( \sqrt[3]{a} \) is the number whose cube equals \( a \). Knowing key square and cube values by heart saves significant time in examinations.
Geometric Interpretation
Squaring a length gives an area; cubing a length gives a volume. This geometric meaning explains why powers appear so frequently in measurement problems.
Key Values to Know
Square numbers and their roots
| \(n\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(n^2\) | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 |
Cube numbers and their roots
| \(n\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| \(n^3\) | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 |
Worked Examples
Squares and Square Roots
\( 13^2 = 13 \times 13 = 169 \), so \( \sqrt{169} = 13 \).
Every positive number has two square roots: positive and negative. \( \sqrt{25} = 5 \) is the principal (positive) root. The full solution to \( x^2 = 25 \) is \( x = \pm 5 \).
Powers of 2: \( 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 \). Each term doubles the previous one.
Cubes and Cube Roots
\( 7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343 \), so \( \sqrt[3]{343} = 7 \).
Unlike square roots, cube roots of negative numbers are real: \( \sqrt[3]{-8} = -2 \) because \( (-2)^3 = -8 \).
Powers of 3: \( 3, 9, 27, 81, 243 \). Powers of 5: \( 5, 25, 125, 625 \).
Higher Powers and Recognising Sequences
Powers of 4: \( 4, 16, 64, 256 \). Note: \( 4^n = (2^2)^n = 2^{2n} \), so powers of 4 appear within the powers-of-2 sequence at every even index.
Mixed expression (apply BIDMAS — indices before addition): \( 2^4 - \sqrt{81} + 3^3 = 16 - 9 + 27 = 34 \).
Key Takeaways
- \( a^n \) means \( a \) multiplied by itself \( n \) times; \( a \) is the base, \( n \) is the index.
- Square roots and cube roots are the inverse operations of squaring and cubing.
- Know square numbers up to \( 15^2 = 225 \) and cube numbers up to \( 10^3 = 1000 \) by heart.
- The principal square root \( \sqrt{a} \) is always positive; the full solution to \( x^2 = a \) is \( x = \pm\sqrt{a} \).
- Cube roots of negative numbers are real; square roots of negative numbers have no real solution.