Indices and Powers Revision Notes for HSC SSCE Mathematics Advanced

Indices and Powers

What are indices?

An index (also called an exponent or power) indicates how many times a number is multiplied by itself. The number being multiplied is called the base.

For example, in the expression $3^4$ , the base is $3$ and the index is $4$ . This means:

$3^4 = 3 \times 3 \times 3 \times 3 = 81$

Understanding indices is essential for simplifying algebraic expressions and solving mathematical problems efficiently.

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The index tells us how many times to use the base as a factor in multiplication. So $3^4$ means "multiply $3$ by itself $4$ times". This is much more efficient than writing out $3 \times 3 \times 3 \times 3$ every time!

The index laws

The index laws are a set of rules that help us simplify expressions involving powers. These laws only apply when the bases are the same.

1. Product rule (multiply → add powers)

When multiplying powers with the same base, we add the indices together.

$a^m \times a^n = a^{m+n}$

Example:

$2^3 \times 2^5 = 2^{3+5} = 2^8 = 256$

This works because we are effectively counting the total number of times the base is multiplied by itself.

2. Quotient rule (divide → subtract powers)

When dividing powers with the same base, we subtract the indices.

$a^m \div a^n = a^{m-n}$

Example:

$5^7 \div 5^3 = 5^{7-3} = 5^4 = 625$

This rule reflects the cancellation of common factors when dividing.

3. Power of a power (multiply powers)

When raising a power to another power, we multiply the indices together.

$(a^m)^n = a^{mn}$

Example:

$(3^2)^4 = 3^{2 \times 4} = 3^8 = 6561$

This is because we are repeatedly multiplying the base by itself multiple times.

4. Power of a product

When raising a product to a power, we distribute the power to each factor in the product.

$(ab)^n = a^n \times b^n$

Example:

$(2 \times 5)^3 = 2^3 \times 5^3 = 8 \times 125 = 1000$

Each factor in the product is raised to the same power separately.

5. Power of a quotient

When raising a quotient to a power, we distribute the power to both the numerator and denominator.

$(a \div b)^n = a^n \div b^n$

This rule is similar to the power of a product, but applies to division.

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Quick memory aid:

Same base + multiply? → Add indices
Same base + divide? → Subtract indices
Power on power? → Multiply indices
Product or quotient with power? → Distribute to each part

Negative indices

A negative index means we take the reciprocal of the base raised to the positive version of that index.

$a^{-n} = \frac{1}{a^n}$

In other words, a negative power "flips" the base into a fraction.

Example:

$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

Negative indices are useful for expressing very small numbers and for simplifying complex algebraic expressions.

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A negative index does NOT make the answer negative! Instead, it creates a fraction. The negative sign tells you to take the reciprocal: $a^{-n} = \frac{1}{a^n}$

Fractional indices

A fractional index represents a root. The denominator of the fraction tells us which root to take, whilst the numerator tells us which power to apply.

$a^{1/n} = \sqrt[n]{a}$

This means " $a$ to the power of $\frac{1}{n}$ " is the same as "the $n$ th root of $a$ ".

For more complex fractional indices:

$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Examples:

$16^{1/2} = \sqrt{16} = 4$

$27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9$

In the second example, we first find the cube root of $27$ (which is $3$ ), then square the result.

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Understanding fractional indices:

The denominator tells you the type of root (2 = square root, 3 = cube root, etc.)
The numerator tells you the power to apply
You can take the root first and then apply the power, or vice versa – the result is the same!

Worked examples

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Worked Example 1: Simplify $3^4 \times 3^{-2}$

Step 1: Same base → add indices

$4 + (-2) = 2$

Answer:

$3^2 = 9$

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Worked Example 2: Multiple laws with $(2^3)^2 \div 2^4$

Step 1: Power of a power

$(2^3)^2 = 2^{3 \times 2} = 2^6$

Step 2: Quotient rule

$2^6 \div 2^4 = 2^{6-4} = 2^2$

Answer:

$= 4$

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Worked Example 3: Fractional and negative indices with $8^{-2/3}$

Step 1: Rewrite base

$8 = 2^3$

Step 2: Multiply powers

$(2^3)^{-2/3} = 2^{3 \times (-2/3)} = 2^{-2}$

Step 3: Negative index → reciprocal

$= \frac{1}{2^2}$

Answer:

$= \frac{1}{4}$

Common mistakes

Students often make errors when working with indices. Here are some common pitfalls to avoid:

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Mistake 1: Trying to add indices when adding terms

$a^m + a^n \neq a^{m+n}$

The product rule only works for multiplication, not addition. The expression $a^m + a^n$ cannot be simplified using index laws.

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Mistake 2: Forgetting to apply negative indices correctly

Remember that $a^{-2} = \frac{1}{a^2}$ , not $-a^2$ or some other variation. The negative sign in the index means take the reciprocal.

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Mistake 3: Misreading fractional indices

$a^{1/2}$ means the square root of $a$ , not $a$ divided by $2$ . The fractional index represents a root operation, not division.

Exam tips

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Strategic approaches for index problems:

Always check if the bases are the same before applying index laws. If the bases differ, you may need to rewrite numbers first.
Rewrite numbers as powers when possible. For example:
- $8 = 2^3$
- $27 = 3^3$
- $16 = 2^4$ or $4^2$
Convert roots into indices to simplify your working. For example, write $\sqrt{x}$ as $x^{1/2}$ .
Work step-by-step in this order:
1. Simplify any powers of powers first
2. Apply the appropriate index laws
3. Evaluate the final expression
In non-calculator questions, rewriting numbers as powers of the same base is often the key step. Look for common bases like $2$ , $3$ , $5$ , or $10$ .

Remember!

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Key Points to Remember:

Indices describe repeated multiplication of a base number.
The index laws help simplify expressions quickly: add indices when multiplying, subtract when dividing, and multiply when raising a power to a power.
Negative indices mean take the reciprocal: $a^{-n} = \frac{1}{a^n}$ .
Fractional indices represent roots: $a^{1/n} = \sqrt[n]{a}$ and $a^{m/n} = \sqrt[n]{a^m}$ .
Always check that bases are the same before applying index laws, and work systematically through problems step by step.

Indices and Powers (HSC SSCE Mathematics Advanced): Revision Notes