Indices in algebra
Indices in algebra
- The index laws work with letters: $a^m \times a^n = a^{m+n}$, $a^m \div a^n = a^{m-n}$, $(a^m)^n = a^{mn}$.
- Examples: $(5x^3)^2 = 25x^6$; $\quad 12a^5 \div 3a^{-2} = 4a^7$; $\quad 6x^7y^4 \times 5x^{-5}y = 30x^2 y^5$.
Practice
(5x³)² = k·x⁶. What is the coefficient k?
(5x³)² = 5² × (x³)² = 25x⁶, so k = 25.
Practice
Simplify 12a⁵ ÷ 3a⁻².
12/3 = 4 and a⁵ ÷ a⁻² = a^(5−(−2)) = a⁷, so 4a⁷.
Index equations
- Write both sides with the same base, then equate the powers.
- Solve $2^x = 32$: since $32 = 2^5$, $x = 5$.
Practice
Solve 2^x = 32.
32 = 2⁵, so x = 5.
You've got it
Key idea
- multiply → add indices; divide → subtract; power of a power → multiply
- $(5x^3)^2 = 25x^6$
- solve index equations by matching the base ($2^x = 32 = 2^5 \Rightarrow x = 5$)