Apply the index laws to expressions with rational indices: use zero, negative and fractional indices, simplify and evaluate index expressions, solve simple index equations, and write numbers in scientific notation

What this dot point is asking

You met the index laws in earlier years for whole-number powers. NESA now expects you to use them with full confidence when the index can be zero, negative or a fraction, to simplify and evaluate index expressions cleanly, to solve simple index equations such as $2^x = 32$ by matching the base, and to handle very large or very small numbers in scientific notation. This is the gateway page for the whole exponential and logarithmic module: a logarithm is just an index in disguise, the curve $y = a^x$ is built from these powers, and the derivative rules in calculus run on the power rule, which is the index laws wearing a hat. The mechanics are short, but they recur in almost every later topic, so the goal is speed without slips and, just as importantly, knowing why each rule holds so you can rebuild it under pressure.

The answer

Power, base and index

The expression $a^n$ is a power. The number $a$ is the base and $n$ is the index (also called the exponent; the words mean exactly the same thing). When $n$ is a positive whole number, the power is just repeated multiplication: $a^n = a \times a \times \dots \times a$ with $n$ factors, so $3^4 = 3 \times 3 \times 3 \times 3 = 81$. Everything that follows extends this single idea to indices that are zero, negative or fractional, and the guiding principle is the one that makes the whole topic coherent: we define the new powers precisely so that the laws you already know keep working.

The five index laws (same base)

Five laws let you combine powers of the same base. The first three combine two powers; the last two push a power across a product or quotient:

• Multiplying adds indices: $a^m \times a^n = a^{m+n}$.
• Dividing subtracts indices: $\dfrac{a^m}{a^n} = a^{m-n}$.
• Power of a power multiplies indices: $\left(a^m\right)^n = a^{mn}$.
• Power of a product distributes: $(ab)^n = a^n b^n$.
• Power of a quotient distributes: $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$.

The first law is obvious once you count factors: $a^2 \times a^3 = (a \times a)(a \times a \times a) = a^5$, and $2 + 3 = 5$. The division law is the same count in reverse, and the power-of-a-power law is repeated use of the first. None of these laws lets you combine different bases: $a^3 \times b^2$ does not simplify, because the bases differ. When an expression has several factors, work through it systematically: the number coefficients first, then each pronumeral in turn. A common and costly slip is to add the coefficients when the law only adds the indices: in $3x^4 \times 4x^3 = 12 x^7$ the indices add ($4 + 3$) but the numbers multiply ($3 \times 4 = 12$).

Zero and negative indices: where the definitions come from

Why is $a^0 = 1$, and why does a negative index mean "take the reciprocal"? Because those are the only values that keep the division law true. Consider $\dfrac{a^3}{a^3}$. By ordinary cancelling it equals $1$. By the division law it equals $a^{3-3} = a^0$. For both to agree we are forced to define $a^0 = 1$. The same argument fixes negative indices: $\dfrac{a^2}{a^3} = \dfrac{1}{a}$ by cancelling, but $a^{2-3} = a^{-1}$ by the law, so we must define $a^{-1} = \dfrac{1}{a}$, and in general $a^{-n} = \dfrac{1}{a^n}$. Both definitions need $a \neq 0$, because they came from dividing by a power of $a$.

So the negative sign in an index says, "take the reciprocal" - and the cleanest habit is to do that first. For a fraction this is especially neat, because the reciprocal just turns the fraction upside down: $\left(\dfrac{2}{3}\right)^{-3} = \left(\dfrac{3}{2}\right)^{3} = \dfrac{27}{8}$. Write the reciprocal of $\dfrac{a}{b}$ as $\dfrac{b}{a}$, not as $\dfrac{1}{a/b}$, and the rest is just a positive power.

Fractional indices: roots and powers

A fractional index brings in roots, and again the definition is chosen to keep the power-of-a-power law alive. Take $a^{1/2}$. The law says $\left(a^{1/2}\right)^2 = a^{(1/2) \times 2} = a^1 = a$, so $a^{1/2}$ is the number that squares to $a$: it is $\sqrt{a}$. The same reasoning gives $a^{1/3} = \sqrt[3]{a}$ and in general $a^{1/n} = \sqrt[n]{a}$, the $n$-th root. Because roots are involved, the base must satisfy $a \ge 0$ for an even root.

A general fraction $\dfrac{m}{n}$ in the index combines a root and a power, and the power-of-a-power law tells you the order does not affect the answer:

In practice take the root first, because it keeps the numbers small. To evaluate $27^{2/3}$, take the cube root before squaring: $\left(\sqrt[3]{27}\right)^2 = 3^2 = 9$. Doing it the other way ($27^2 = 729$, then $\sqrt[3]{729} = 9$) gives the same value but with far heavier arithmetic.

The order for a messy index: reciprocal, then root, then power

When an index is both negative and fractional, deal with the pieces in a fixed order and each step stays simple. This single routine handles every numerical index question in the course, and it is the kind of explicit method the textbook leaves you to assemble yourself.

Solving simple index equations

An equation such as $2^x = 32$ asks "what power of $2$ gives $32$?". The reliable method at this level is to write both sides as powers of the same base and then equate the indices, because a power is completely determined by its index once the base is fixed. Since $32 = 2^5$, the equation $2^x = 2^5$ forces $x = 5$. The same trick handles equations where the two sides use different-looking bases that are secretly powers of one common base: for $8^x = 4$, write $8 = 2^3$ and $4 = 2^2$, so $\left(2^3\right)^x = 2^2$ gives $2^{3x} = 2^2$ and hence $3x = 2$, that is $x = \dfrac{2}{3}$. (When the two sides cannot be matched to a common base, you need logarithms - the very next page in this module.)

Scientific notation

Scientific notation writes a number as $a \times 10^k$, where $1 \le a < 10$ (one non-zero digit before the decimal point) and $k$ is an integer. It is the natural home for the index laws, because multiplying and dividing such numbers just means handling the number parts and the powers of ten separately. To multiply, multiply the $a$ values and add the indices of $10$; to divide, divide the $a$ values and subtract the indices. After combining, you may need to renormalise so the front number again lies between $1$ and $10$: for instance $12 \times 10^8 = 1.2 \times 10^1 \times 10^8 = 1.2 \times 10^9$.

The growth curve below is the simplest exponential, $y = 2^x$, the same doubling that runs through index equations like $2^x = 32$. Reading values off it shows the index laws at work: each step of $1$ to the right doubles $y$ (multiplying by $2^1$), each step left halves it (the negative-index reciprocal), and the value at $x = 0$ is $2^0 = 1$.

How exam questions ask about indices

A handful of command words tell you exactly which skill is wanted:

• "Simplify" an index expression means combine the powers using the laws and present the answer in index form. Unless told otherwise, leave no negative or zero index in the final form, and remember to multiply the number coefficients.
• "Simplify, leaving your answer with positive indices" is the explicit version of the same instruction: any $a^{-n}$ must be rewritten as $\dfrac{1}{a^n}$.
• "Evaluate" or "find the value of" a numerical power means produce an exact number. For a fractional index, take the root first; for a negative index, take the reciprocal first.
• "Solve" an index equation such as $2^x = 32$ means write both sides as powers of one base and equate the indices.
• "Express in scientific notation" signals an answer of the form $a \times 10^k$ with $1 \le a < 10$; renormalise the front number if a calculation leaves it outside that range.
• "Express in index form" or "using a fractional index" asks you to rewrite a root, such as $\sqrt[3]{x^2} = x^{2/3}$, ready for the index laws or for differentiation later.