Indices (Edexcel GCSE Maths): Revision Notes

Worked Example: Multiplying Powers

Calculate: $4^3 × 4^7$

Step 1: Identify that both terms have the same base (4) Step 2: Add the indices: $3 + 7 = 10$ Step 3: Write the answer: $4^3 × 4^7 = 4^{10}$

Worked Example: Dividing Powers

Calculate: $12^6 ÷ 12^3$

Step 1: Identify that both terms have the same base (12) Step 2: Subtract the indices: $6 - 3 = 3$ Step 3: Write the answer: $12^6 ÷ 12^3 = 12^3$

Worked Example: Power of a Power

Calculate: $(7^3)^5$

Step 1: Identify the inner power (3) and outer power (5) Step 2: Multiply the indices: $3 × 5 = 15$ Step 3: Write the answer: $(7^3)^5 = 7^{15}$

Worked Example: Negative Powers

Calculate: $5^{-2}$

Step 1: Apply the negative power rule Step 2: $5^{-2} = \frac{1}{5^2}$ Step 3: Simplify: $\frac{1}{5^2} = \frac{1}{25}$

Worked Example: Reciprocals

Calculate: $\left(\frac{5}{9}\right)^{-1}$

Step 1: Apply the reciprocal rule for fractions Step 2: Flip the fraction: $\left(\frac{5}{9}\right)^{-1} = \frac{9}{5}$

Worked Example: Powers of Fractions

Calculate: $\left(\frac{3}{10}\right)^2$

Step 1: Apply the power to both top and bottom Step 2: $\left(\frac{3}{10}\right)^2 = \frac{3^2}{10^2}$ Step 3: Simplify: $\frac{3^2}{10^2} = \frac{9}{100}$

Worked Example: Combining Rules

Calculate: $\left(\frac{2}{3}\right)^{-3}$

Step 1: Apply the negative power rule (flip the fraction) Step 2: $\left(\frac{2}{3}\right)^{-3} = \left(\frac{3}{2}\right)^3$ Step 3: Apply the power to numerator and denominator Step 4: $\left(\frac{3}{2}\right)^3 = \frac{3^3}{2^3} = \frac{27}{8}$

Worked Example: Complex Simplification

Question: Simplify $\frac{3^7 × 3}{3^4}$ fully, leaving your answer in index form.

Solution: Step 1: Use the rule $a^m × a^n = a^{(m+n)}$ to simplify the numerator

• $3^7 × 3 = 3^7 × 3^1 = 3^8$

Step 2: Use the rule $\frac{a^m}{a^n} = a^{(m-n)}$ to simplify the fraction

• $\frac{3^8}{3^4} = 3^{(8-4)} = 3^4$

Answer: $3^4$