Algebraic Roots & IndicesCambridge IGCSE Maths: Flashcards

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State the index law for $a^m \times a^n$

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State the index law for $a^m \times a^n$: $a^m \times a^n = a^{m+n}$
State the index law for $a^m \div a^n$: $a^m \div a^n = a^{m-n}$
State the index law for $(a^m)^n$: $(a^m)^n = a^{mn}$
What does $a^0$ equal (for non-zero $a$ )?: $1$
What does a negative index mean, e.g. $a^{-n}$ ?: The reciprocal of the positive-index term: $a^{-n} = \frac{1}{a^n}$
What does a fractional index $a^{\frac{1}{n}}$ mean?: The $n$ th root of $a$ : $a^{\frac{1}{n}} = \sqrt[n]{a}$
Evaluate $16^{\frac{1}{4}}$: $\sqrt[4]{16} = 2$
Simplify $(5x^3)^2$: $25x^6$
Simplify $12a^5 \div 3a^{-2}$: $4a^7$
Simplify $6x^7y^4 \times 5x^{-5}y$: $30x^2y^5$
Solve $2^x = 32$: $x = 5$ , since $32 = 2^5$
Solve $5^{x+1} = 25^x$: $x = 1$ , since $25^x = 5^{2x}$ gives $x+1=2x$
What is the strategy for solving equations where the unknown is in the index?: Rewrite both sides using the same base, then equate the indices.
Is knowledge of logarithms required for solving index equations at IGCSE?: No, logarithms are not required — same-base equating is always sufficient.