Laws of Indices

In your AQA exam, you will usually be asked to simplify an algebraic expression by writing it in its most compact index form, or to evaluate a numerical expression by finding its actual numerical value.

The Multiplication Law

• The Rule: $a^m \times a^n = a^{m+n}$
• When multiplying terms with the exact same base, you add the indices together.
• If the terms have numerical coefficients, you multiply the coefficients normally and then add the indices of the matching variables.
• Any variable written without an index has a hidden power of $1$ (e.g., $x = x^1$).

Worked Example 1: Multiplication with Algebraic Bases

Simplify $4a^3 \times 7a^2$

Step 1: Multiply the numerical coefficients.

• $4 \times 7 = 28$

Step 2: Add the indices for the matching algebraic bases.

• $3 + 2 = 5$

Step 3: Combine into the final simplified expression.

• Result: $28a^5$

Worked Example 2: Multiplication with Numerical Bases

Simplify $6^3 \times 6^4$

Step 1: Check that the bases are exactly the same.

• Both bases are $6$.

Step 2: Add the indices together.

• $3 + 4 = 7$

Step 3: Write the final simplified expression.

• Result: $6^7$

The Division Law

• The Rule: $a^m \div a^n = a^{m-n}$
• When dividing terms with the same base, subtract the index of the divisor (the second term) from the index of the dividend (the first term).
• This is often written as an algebraic fraction: $\frac{a^m}{a^n}$.
• Divide the coefficients first, then subtract the indices of the matching bases.

Worked Example 1: Division with Algebraic Bases

Simplify $18a^{-6} \div 9a^{-4}$

Step 1: Divide the numerical coefficients.

• $18 \div 9 = 2$

Step 2: Subtract the indices, being very careful with negative numbers.

• $-6 - (-4) = -6 + 4 = -2$

Step 3: Combine into the final expression.

• Result: $2a^{-2}$ (which can also be written using a reciprocal as $\frac{2}{a^2}$)

Worked Example 2: Division with Numerical Bases

Simplify $5^8 \div 5^2$

Step 1: Check that the bases are exactly the same.

• Both bases are $5$.

Step 2: Subtract the index of the divisor from the index of the dividend.

• $8 - 2 = 6$

Step 3: Write the final simplified expression.

• Result: $5^6$

The Power of a Power Law

• The Rule: $(a^m)^n = a^{m \times n}$
• When a base that already has a power is raised to another power, you multiply the indices together.
• If there is a coefficient inside the bracket, the power outside the bracket must be applied to that coefficient as well: $(nx^a)^b = n^b x^{ab}$.

Worked Example 1: Power of a Power with Algebraic Bases

Simplify $(3x^2)^4$

Step 1: Apply the outside power to the coefficient.

• $3^4 = 3 \times 3 \times 3 \times 3 = 81$

Step 2: Multiply the indices for the variable.

• $2 \times 4 = 8$

Step 3: Combine into the final expression.

• Result: $81x^8$

Worked Example 2: Power of a Power with Numerical Bases

Simplify $(4^2)^3$

Step 1: Multiply the inside index by the outside index.

• $2 \times 3 = 6$

Step 2: Write the final simplified expression keeping the same base.

• Result: $4^6$

Sums, Subtractions, and Collecting Like Terms

• There is absolutely no index law for addition or subtraction. You cannot simplify $x^2 + x^3$ into a single term.
• You can only add or subtract terms if they are like terms, meaning they have the identical base AND the identical index.
• When collecting like terms, you only add or subtract the coefficients. The base and the index remain completely unchanged.

Worked Example: Adding and Subtracting Indices

Simplify $5a^3 - 2a^2 + 4a^3 + 7a^2$

Step 1: Group the terms that share the exact same base and index.

• Group 1: $5a^3$ and $+4a^3$
• Group 2: $-2a^2$ and $+7a^2$

Step 2: Add or subtract the coefficients for each group separately.

• $5 + 4 = 9a^3$
• $-2 + 7 = +5a^2$

Step 3: Write the final simplified expression.

• Result: $9a^3 + 5a^2$