How to use the laws of indices

An index (or exponent) is the small raised number that tells you how many times a base is multiplied by itself, as in $2^{5}=2\times2\times2\times2\times2$. The laws of indices are the rules for combining such powers without writing them out in full.

In Add Math you use them to simplify expressions, to write a messy product or quotient as a single power, and, crucially, to rewrite different bases as powers of one common base before solving an index equation.

Three core laws do most of the work, and three special cases handle the awkward exponents.

The special cases: $a^{0}=1$, $a^{-n}=\dfrac{1}{a^{n}}$, and a fractional index means a root, $a^{\frac{m}{n}}=\sqrt[n]{a^{m}}$.

Reach for the laws whenever you see powers of the same base being multiplied, divided or raised again, or whenever an expression mixes numbers like $4$, $8$ and $16$ that are all powers of a smaller base. Instructions such as 'simplify', 'express as a single power of $2$', or 'evaluate' point straight to them.

They are also the first move in almost every index equation. If a question reads $9^{x}=27$, you cannot compare the exponents until both sides share a base, here $3$.

Rewriting $9=3^{2}$ and $27=3^{3}$ turns the equation into $3^{2x}=3^{3}$, and then the powers can be equated. Spotting the common base early is what keeps the working short and every line easy to credit.

1. 1

   Choose a common base

   Rewrite every number as a power of the same base where possible, for example $8=2^{3}$, $4=2^{2}$, $16=2^{4}$.

2. 2

   Clear the brackets

   Use $(a^{m})^{n}=a^{mn}$ to remove any power of a power, so each factor is a single base to a single index.

3. 3

   Multiply like bases

   Where the same base is multiplied, add the indices with $a^{m}\times a^{n}=a^{m+n}$.

4. 4

   Divide like bases

   Where the same base is divided, subtract the indices with $a^{m}\div a^{n}=a^{m-n}$.

5. 5

   Tidy special indices

   Apply $a^{0}=1$, turn any negative index into a reciprocal with $a^{-n}=\frac{1}{a^{n}}$, and read fractional indices as roots.

6. 6

   State the single power

   Collect everything into one power of the base, and evaluate it if the question asks for a number.

• Multiplying the indices when bases are multiplied, $a^{m}\times a^{n}$ is $a^{m+n}$, you add, not multiply.
• Adding the indices when raising a power to a power, $(a^{m})^{n}$ is $a^{mn}$, you multiply here.
• Trying to combine different bases directly, such as $2^{3}\times 3^{2}$; the laws only apply when the base is the same.
• Reading $a^{-n}$ as a negative number instead of a reciprocal, $2^{-3}=\frac{1}{8}$, which is positive.
• Forgetting $a^{0}=1$, or dropping the exponent when it becomes zero after subtraction.

The slip we see most is a student multiplying indices when the bases are merely multiplied, or forgetting to rewrite $8$, $4$ and $16$ as powers of $2$ before starting. In one-to-one lessons our teachers make you name which law you are using on every line, so 'add for times, subtract for divide, multiply for a power' becomes automatic rather than guessed.

We also drill the common-base step on numbers like $9=3^{2}$ and $27=3^{3}$, because that single move unlocks almost every index equation. Every teacher on spmaddmath.com.my is experienced.

Lessons are online and taught in English, paced to how quickly the rules are clicking for you.