Indices and the laws of exponents (QCE Math Methods Unit 2)

What this dot point is asking

QCAA wants you to apply the laws of indices with confidence, including with rational and negative exponents, and to use the laws to solve exponential equations where both sides can be rewritten with the same base.

The index laws

For any positive base $a$ and rationals $m$, $n$:

Rational exponents: $a^{1/n} = \sqrt[n]{a}$ and $a^{m/n} = (\sqrt[n]{a})^m$.

These laws come directly from the definition of repeated multiplication and extend smoothly to negative and rational powers.

Common manipulations

The recurring skill is to rewrite every term as a power of one base before combining.

Negative exponent

$5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}$, moving the power to the denominator.

Rational exponent

$8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$, taking the cube root then squaring.

Same base products

$2^x \cdot 2^{3x + 1} = 2^{4x + 1}$, adding exponents.

Rebasing

$9^{x + 1} = (3^2)^{x + 1} = 3^{2x + 2}$, which lets a base-$9$ term combine with base-$3$ terms.

These manipulations are also the gateway to differentiating and integrating power functions, where an expression like $\dfrac{1}{\sqrt x}$ must first be rewritten as $x^{-1/2}$ before the power rule can apply.

Recognising base families

The same-base method depends on spotting that several numbers are powers of one base. The powers of $2$ are $2, 4, 8, 16, 32, \ldots$; the powers of $3$ are $3, 9, 27, 81, \ldots$; the powers of $5$ are $5, 25, 125, \ldots$. When a question mixes, say, $4$ and $8$ and $16$, every term can be written with base $2$, after which the laws combine them and the exponents can be equated. Building familiarity with these families makes most Paper 1 index questions immediate, because the rewrite is the only hard step.

Solving exponential equations (same-base method)

If both sides of an equation can be rewritten with the same base, equate exponents and solve. This works when the bases are related by integer or rational powers (for example $4, 8, 16$ all relate to base $2$; $9, 27, 81$ all relate to base $3$). The justification is that the exponential function is one-to-one: if $a^p = a^q$ for a fixed base $a > 0$, $a \neq 1$, then $p = q$, so matching the powers is valid.

When the bases cannot be aligned (for example $2^x = 7$), logarithms are required (the next dot point covers this).

Why the laws extend to all exponents

The laws originate in repeated multiplication: $a^3 \cdot a^2 = (a \cdot a \cdot a)(a \cdot a) = a^5$. To keep the rule $a^m / a^n = a^{m-n}$ working when $m = n$ forces $a^0 = 1$, and when $m < n$ it forces $a^{-k} = \tfrac{1}{a^k}$. Likewise, requiring $(a^{1/n})^n = a^{1} = a$ forces $a^{1/n} = \sqrt[n]{a}$. So the negative and fractional cases are not new rules but the only definitions that keep the original laws consistent, which is why they can be applied with confidence.

How this appears in IA1 and EA

IA1

Direct application questions: simplify a multi-term expression with mixed positive, negative and fractional exponents, or solve a same-base equation.

EA Paper 1

Multiple choice on index manipulations and short solve problems.

EA Paper 2

Used as the algebra step inside a larger problem (an exponential growth model, a calculus-of-exponentials computation).