Powers and Roots

What you will learn

• Read and use powers, such as $2^5$, and roots.
• Use the index laws for multiplying and dividing powers.
• Write large and small numbers in standard form.

Before you start: you should be comfortable with multiplication and order of operations. See Order of Operations.

The idea

Writing $2 \times 2 \times 2 \times 2 \times 2$ gets tedious, so we write $2^5$ instead. The $2$ is the base and the small raised $5$ is the index (also called the power or exponent). It tells you how many copies of the base to multiply together. $2^5 = 32$.

• $a^2$ is ”$a$ squared”, the area of a square with side $a$.
• $a^3$ is ”$a$ cubed”, the volume of a cube with side $a$.

A root undoes a power. The square root $\sqrt{49} = 7$ because $7^2 = 49$. The cube root $\sqrt[3]{64} = 4$ because $4^3 = 64$.

Why the index laws work

Multiply $a^3 \times a^2$ and write it out: $(a \times a \times a) \times (a \times a)$. That is five $a$‘s, so $a^3 \times a^2 = a^5$. You add the indices. Every other law follows from the same “count the copies” reasoning.

Zero and negative powers

Follow the pattern $2^3 = 8$, $2^2 = 4$, $2^1 = 2$. Each step down divides by $2$. Continue: $2^0 = 1$, $2^{-1} = \frac{1}{2}$, $2^{-2} = \frac{1}{4}$. That is why any non-zero number to the power $0$ is $1$, and a negative power means “one over”.

Fractional powers

Since $(a^{1/2})^2 = a^1 = a$, the power $\frac{1}{2}$ must mean square root. Likewise $a^{1/3} = \sqrt[3]{a}$. A power like $a^{2/3}$ means “cube root, then square”.

Worked examples

Example 1: Simplifying with index laws

Simplify $\dfrac{x^7 \times x^4}{x^5}$.

1. Multiply on top by adding indices: $x^7 \times x^4 = x^{11}$.
2. Divide by subtracting indices: $x^{11} \div x^5 = x^{6}$.

Answer: $x^6$

Example 2: Evaluating a fractional power

Work out $27^{2/3}$ without a calculator.

1. The denominator $3$ means take the cube root: $\sqrt[3]{27} = 3$.
2. The numerator $2$ means square the result: $3^2 = 9$.

Answer: $9$

Example 3: Standard form

The distance to the Sun is about $150\,000\,000$ km. Write this in standard form.

1. Place the decimal point after the first non-zero digit: $1.5$.
2. Count how many places the point moved: $8$.
3. Write as $1.5 \times 10^8$ km.

Answer: $1.5 \times 10^{8}$ km

$(-3)^2 = 9$ but $-3^2 = -9$. The power applies only to what it touches, so brackets matter.

Why it matters

Powers describe growth that multiplies rather than adds: bacteria doubling every hour, compound interest, computer memory ($2^{10} = 1024$ bytes in a kibibyte) and the energy of an earthquake on the Richter scale. Roots appear whenever we go the other way, for example finding a side length from an area or a speed from kinetic energy.

Standard form, built on powers of $10$, lets scientists write the mass of an electron ($9.11 \times 10^{-31}$ kg) and the size of the observable universe on the same page.