Laws of Indices

Welcome, students 👋 In this lesson, you will learn one of the most useful tools in algebra: the laws of indices. These rules let us simplify expressions with powers quickly and accurately. They appear everywhere in mathematics, from scientific notation to compound growth, and they are essential for IB Mathematics: Analysis and Approaches HL.

What are indices and why do they matter?

An index tells you how many times a number is multiplied by itself. For example, $2^4$ means $2 \times 2 \times 2 \times 2$. Here, $2$ is the base and $4$ is the index or exponent.

Indices are important because they make repeated multiplication easier to write and easier to work with. Instead of writing long products, we can use compact forms. This is especially helpful in Number and Algebra, where we often need to simplify expressions, solve equations, and model patterns.

The laws of indices describe how powers behave when bases are the same. If you know these rules, you can simplify expressions such as $x^3 \cdot x^5$, $\left(y^2\right)^4$, or $\dfrac{a^7}{a^2}$ without expanding everything fully. That saves time and reduces mistakes ✅

Key vocabulary

$a$ is the base
$n$ is the exponent or index
A power is written as $a^n$
A product means multiplication
A quotient means division

These terms will come up often, so students, make sure they feel familiar.

The main laws of indices

The first law says that when you multiply powers with the same base, you add the indices:

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

This works because repeated multiplication combines. For example,

$$x^2 \cdot x^3 = \left(x \cdot x\right) \left(x \cdot x \cdot x\right) = x^5$$

The second law says that when you divide powers with the same base, you subtract the indices:

$$\frac{a^m}{a^n} = a^{m-n}, \quad a \neq 0$$

For example,

$$\frac{y^7}{y^4} = y^3$$

because three $y$ factors remain after canceling four from top and bottom.

The third law says that when you raise a power to another power, you multiply the indices:

$$\left(a^m\right)^n = a^{mn}$$

For example,

$$\left(z^2\right)^3 = z^6$$

because $\left(z^2\right)^3$ means $z^2 \cdot z^2 \cdot z^2$, which gives six $z$ factors.

The fourth law says that a power of a product is the product of the powers:

$$\left(ab\right)^n = a^n b^n$$

For example,

$$\left(2x\right)^3 = 2^3 x^3 = 8x^3$$

This is very useful when simplifying expressions with brackets.

The fifth law says that a power of a quotient is the quotient of the powers:

$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}, \quad b \neq 0$$

For example,

$$\left(\frac{3x}{2}\right)^2 = \frac{9x^2}{4}$$

These five rules are the foundation of the topic. They are used again and again in algebra, equations, graphs, and scientific notation.

Zero, negative, and fractional indices

The laws above also help explain special indices. These are important in IB Mathematics: Analysis and Approaches HL because they extend the rules beyond whole numbers.

Zero index

Any non-zero number raised to the power $0$ equals $1$:

$$a^0 = 1, \quad a \neq 0$$

Why? Use the division law:

$$\frac{a^m}{a^m} = a^{m-m} = a^0$$

But any non-zero number divided by itself is $1$, so $a^0 = 1$.

Example:

$$5^0 = 1$$

Negative indices

A negative index means the reciprocal:

$$a^{-n} = \frac{1}{a^n}, \quad a \neq 0$$

This comes from the division law too. For example,

$$\frac{a^3}{a^5} = a^{-2}$$

and also

$$\frac{a^3}{a^5} = \frac{1}{a^2}$$

so $a^{-2} = \frac{1}{a^2}$.

Example:

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

Fractional indices

Fractional indices connect powers and roots:

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

and more generally,

$$a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$$

Example:

$$16^{\frac{1}{2}} = 4$$

because $\sqrt{16} = 4$.

Another example:

$$27^{\frac{2}{3}} = \left(\sqrt[3]{27}\right)^2 = 3^2 = 9$$

These ideas are especially useful when simplifying algebraic expressions and solving equations with powers.

How to simplify expressions correctly

A common IB skill is simplifying expressions by applying several index laws in sequence. Always look for the same base, powers of powers, and bracketed terms.

Example 1:

$$\frac{x^5 \cdot x^2}{x^3}$$

First combine the numerator:

$$x^5 \cdot x^2 = x^7$$

Then divide:

$$\frac{x^7}{x^3} = x^4$$

So the simplified form is $x^4$.

Example 2:

$$\left(3a^2b\right)^2$$

Apply the power to each factor:

$$3^2 \cdot \left(a^2\right)^2 \cdot b^2 = 9a^4b^2$$

Example 3:

$$\frac{4x^{-2}y^3}{2xy^{-1}}$$

Start with coefficients:

$$\frac{4}{2} = 2$$

Then subtract indices for matching bases:

$$x^{-2}/x^1 = x^{-3}$$

and

$$y^3/y^{-1} = y^4$$

So the expression becomes

$$2x^{-3}y^4$$

You may also rewrite it without negative indices:

$$\frac{2y^4}{x^3}$$

That form is often preferred in final answers.

Common mistake alert 🚫

A very common error is to think

$$x^2 + x^3 = x^5$$

This is false. The addition law does not exist for indices. You can only combine powers by adding exponents when the factors are multiplied, not when they are added.

So:

$$x^2 + x^3$$

cannot be simplified further unless you factor:

$$x^2\left(1+x\right)$$

Applying laws of indices in IB Mathematics: Analysis and Approaches HL

In HL work, laws of indices appear in more advanced contexts, not just basic simplification. They support algebraic proof, exponent equations, logarithms, and modeling real situations.

Solving equations with indices

Suppose you need to solve

$$2^{x+1} = 16$$

Rewrite $16$ as a power of $2$:

$$16 = 2^4$$

Then

$$2^{x+1} = 2^4$$

Since the bases are equal, the indices are equal:

$$x+1 = 4$$

$$x = 3$$

This reasoning is very important in algebra.

Exponential growth and decay

Indices also model repeated multiplication in real life, such as population growth, compound interest, and radioactive decay. A quantity that grows by the same factor each period can be written using powers.

For example, if a bank account increases by a factor of $1.05$ each year, then after $n$ years the amount is

$$A = P\left(1.05\right)^n$$

where $P$ is the initial amount.

This is not just a formula to memorize. It is a way of describing repeated multiplication over time.

Links to the wider Number and Algebra topic

The laws of indices connect directly to many other parts of Number and Algebra:

They help simplify algebraic expressions before solving equations.
They support work with sequences and series, especially geometric sequences.
They are useful in scientific notation, where very large or very small numbers are written using powers of $10$.
They prepare you for logarithms, because logarithms are the inverse operation of exponentiation.

For example, a geometric sequence such as

$$3,\ 6,\ 12,\ 24,\dots$$

has each term multiplied by $2$. Its general term can be written using indices:

$$u_n = 3 \cdot 2^{n-1}$$

This shows how index laws help describe patterns efficiently.

Conclusion

The laws of indices are a core part of algebra. They allow you to simplify products, quotients, powers of powers, and expressions with zero, negative, and fractional indices. For students, mastering these rules builds a strong foundation for the rest of IB Mathematics: Analysis and Approaches HL.

When you use the laws correctly, you can work faster, write cleaner solutions, and solve more advanced problems with confidence. These rules are not isolated facts; they connect to equations, sequences, modeling, scientific notation, and logarithms. In short, laws of indices are a central language of Number and Algebra 📘

Study Notes

An index shows repeated multiplication, such as $2^4 = 2 \times 2 \times 2 \times 2$.
Multiply same bases by adding exponents: $a^m \cdot a^n = a^{m+n}$.
Divide same bases by subtracting exponents: $\dfrac{a^m}{a^n} = a^{m-n}$, with $a \neq 0$.
Power of a power means multiply exponents: $\left(a^m\right)^n = a^{mn}$.
Power of a product: $\left(ab\right)^n = a^n b^n$.
Power of a quotient: $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$, with $b \neq 0$.
Zero index rule: $a^0 = 1$, with $a \neq 0$.
Negative index rule: $a^{-n} = \dfrac{1}{a^n}$.
Fractional indices connect to roots: $a^{\frac{1}{n}} = \sqrt[n]{a}$.
Do not add exponents when terms are added; only multiply or divide powers using the index laws.
These rules are essential for simplifying expressions, solving equations, and working with exponential models in IB Mathematics: Analysis and Approaches HL.