Indices flashcards

Key points

• The index, the small raised number, says how many copies of the base to multiply together.
• 3⁴ = 3 × 3 × 3 × 3 = 81: four 3s multiplied, not 3 × 4.
• x² is "x squared" and x³ is "x cubed".

Example

Key points

• A root undoes a power: $\sqrt{81}$ = 9 because 9² = 81.
• The cube root of 64 is 4, because 4³ = 4 × 4 × 4 = 64.
• Learn the squares up to 15² = 225 and the cubes up to 5³ = 125.

Example

Key points

• To multiply powers of the same base, add the indices: aᵐ × aⁿ = aᵐ⁺ⁿ.
• To divide powers of the same base, subtract the indices: aᵐ ÷ aⁿ = aᵐ⁻ⁿ.
• It works because 2³ × 2⁵ is three 2s and then five more: eight 2s.

Example

Key points

• For a power of a power, multiply the indices: (aᵐ)ⁿ = aᵐⁿ.
• (2³)⁴ is four lots of 2³, which is twelve 2s: 2¹².
• The laws only work when the bases are the same: 2³ × 5² cannot be one power.

Example

Key points

• Any number except 0 to the power 0 is 1: 7⁰ = 1.
• Why: 7² ÷ 7² = 7⁰ by the index law, and 49 ÷ 49 = 1.
• In a bigger expression, only the base takes the 0: 3 × 4⁰ = 3 × 1 = 3.

Example

Key points

• A negative index means one over: a⁻ⁿ = $\frac{1}{a^n}$.
• So 2⁻³ = $\frac{1}{2^3}$ = $\frac{1}{8}$, and 4⁻² = $\frac{1}{16}$.
• On a fraction, the minus flips it: ${\left(\frac{3}{4}\right)}^{-1}$ = $\frac{4}{3}$.
• The negative sign never makes the answer negative.

Example

Key points

• A power of $\frac{1}{2}$ is a square root: ${25}^{\frac{1}{2}}$ = $\sqrt{25}$ = 5.
• A power of $\frac{1}{3}$ is a cube root, and a power of $\frac{1}{n}$ is an nth root.
• Why: ${25}^{\frac{1}{2}}$ × ${25}^{\frac{1}{2}}$ = 25¹ = 25, so ${25}^{\frac{1}{2}}$ is the number that squares to 25.

Example

Key points

• The bottom of the fraction is the root; the top is the power.
• Take the root first: it keeps the numbers small.
• So ${27}^{\frac{2}{3}}$ means cube-root 27, then square the answer.

Example

Key points

• Deal with each part in turn: the minus means one over, the bottom is the root, the top is the power.
• Work out the positive power first, then write one over it.
• On a fraction, flip it first: ${\left(\frac{4}{9}\right)}^{\frac{-1}{2}}$ = ${\left(\frac{9}{4}\right)}^{\frac{1}{2}}$.

Example

Key points

• Deal with the numbers and the letters separately.
• Multiplying: multiply the numbers and add the indices.
• Dividing: divide the numbers and subtract the indices.

Example

Key points

• Everything inside the bracket takes the power: the number as well as each letter.
• Raise the number to the power, and multiply each letter's index by the power.
• (2x³)² means (2x³) × (2x³), which is 4x⁶.

Example