Laws of indices: rules for powers with examples

The multiplying and dividing laws only work when the base is the same: $2^3 \times 3^2$ can't be simplified this way.

Divide a power by itself: $\tfrac{a^3}{a^3} = 1$, but the dividing law gives $a^{3-3} = a^0$. So $a^0$ must be 1.

Now divide by a bigger power: $\tfrac{a^2}{a^5} = \tfrac{1}{a^3}$, and the law gives $a^{2-5} = a^{-3}$. So $a^{-3} = \tfrac{1}{a^3}$. A negative power doesn't make the answer negative; it makes it a reciprocal.

Deal with each part of the power separately: the denominator is a root, the numerator is a power, and a minus sign means reciprocal.

Example: $16^{-\frac{3}{4}}$

1. Root: $\sqrt[4]{16} = 2$.
2. Power: $2^3 = 8$.
3. Reciprocal: $\tfrac{1}{8}$.

Taking the root first keeps the numbers small. For a fraction base, flip it for the negative power: $\left(\tfrac{2}{3}\right)^{-2} = \left(\tfrac{3}{2}\right)^2 = \tfrac{9}{4}$.

Deal with the numbers and the letters separately:

• $3x^2 \times 4x^5 = 12x^7$ (multiply 3 by 4, add the powers).
• $12x^8 \div 4x^3 = 3x^5$ (divide 12 by 4, subtract the powers).
• $(2x^3)^4 = 2^4 x^{12} = 16x^{12}$ (the power applies to the number too).
• $\sqrt{x} = x^{\frac{1}{2}}$ and $\tfrac{1}{x^2} = x^{-2}$: useful in differentiation.

Write both sides with the same base, then set the powers equal.

• $2^x = 32$: since $32 = 2^5$, $x = 5$.
• $9^x = 27$: write both as powers of 3: $3^{2x} = 3^3$, so $2x = 3$ and $x = \tfrac{3}{2}$.

When there's no common base, you need logarithms.

• Multiplying the powers when you should add: $x^3 \times x^4 = x^7$, not $x^{12}$.
• Thinking $a^0 = 0$. It's 1.
• Thinking a negative power gives a negative answer: $2^{-3} = \tfrac{1}{8}$.
• Forgetting the number in brackets: $(3x)^2 = 9x^2$, not $3x^2$.
• Order matters with negatives: $(-3)^2 = 9$, but $-3^2 = -9$.

1. Simplify $a^5 \times a^3 \div a^2$.
2. Evaluate $25^{\frac{1}{2}}$ and $25^{-\frac{1}{2}}$.
3. Evaluate $32^{\frac{3}{5}}$.
4. Simplify $(5y^2)^3$.
5. Solve $4^x = 64$.

Answers: 1. $a^6$. 2. 5 and $\tfrac{1}{5}$. 3. $\sqrt[5]{32} = 2$, $2^3 = 8$. 4. $125y^6$. 5. $x = 3$.

More practice in our laws of indices questions.

• On non-calculator papers, learn the first few powers of 2, 3, 4 and 5 and the cube numbers up to $10^3$: fractional index questions use them.
• Show each step of a negative fractional power (root, power, reciprocal) for method marks.
• Check your answer is sensible: a negative power of a whole number greater than 1 gives an answer between 0 and 1.