Recall and apply the laws of indices to simplify expressions and solve equations involving rational and negative exponents

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 $m$, $n$:

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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

Negative exponent. $5^{-2} = 1/25$.

Rational exponent. $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$.

Same base. $2^x \cdot 2^{3x+1} = 2^{4x+1}$.

Different bases, rewritten. $9^{x+1} = (3^2)^{x+1} = 3^{2x+2}$, so $9^{x+1} = 3 \cdot 3^{2x+1}$.

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 $3$).

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

Worked example

Simplify $\dfrac{(2x)^3 \cdot x^{-2}}{4x^{1/2}}$.

Numerator: $(2x)^3 = 8x^3$. So $(2x)^3 \cdot x^{-2} = 8 x^{3-2} = 8 x$.

Divide: $\dfrac{8x}{4 x^{1/2}} = 2 x^{1 - 1/2} = 2 x^{1/2} = 2\sqrt{x}$.

Common traps

Adding exponents on different bases. $2^3 \cdot 3^2$ is not $6^5$. Index laws only work when bases match.

Misapplying the negative exponent. $-2^4$ is $-16$, not $16$. The negative is outside the power. $(-2)^4 = 16$.

Forgetting to align bases. $4^x = 32$ becomes $2^{2x} = 2^5$, so $x = 5/2$.

Treating $0^0$ as $1$. QCAA does not test $0^0$. Avoid it; if it appears, treat it as undefined.

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).

In one sentence

Indices obey seven laws ($a^m a^n = a^{m+n}$, $a^m / a^n = a^{m-n}$, $(a^m)^n = a^{mn}$, product and quotient over different bases, $a^0 = 1$, $a^{-n} = 1/a^n$, rational exponents), and any exponential equation whose two sides can be written with the same base reduces to equating the exponents.