Indices: powers, bases and notation

When you write $2^3$, you don't mean two times three. You mean two multiplied by itself three times: $2 \times 2 \times 2 = 8$. The small $3$ above the $2$ is the index (plural: indices), and it tells you how many times to multiply the base by itself.

Indices are everywhere in mathematics — in algebra, in scientific notation, in compound interest, in geometric sequences. WAEC tests indices directly in every paper, both in objective and theory sections. Without solid foundations in this topic, indicial equations, logarithms, and surds will all feel impossible.

This lesson defines the basic terms, shows how to read and write powers, computes simple powers by hand, and introduces the special cases ($0^1$, $1^n$, etc.) that will return throughout the topic.

Learning objectives

By the end of this lesson you should be able to:

• read and write numbers in index form
• identify the base and the index of a power
• evaluate powers, including powers of negative numbers and fractions
• recognise powers in patterns and sequences

Reading and writing powers

• $2^3$ = "two to the power of three" or "two cubed" = $2 \times 2 \times 2 = 8$
• $5^2$ = "five squared" = $5 \times 5 = 25$
• $10^4$ = "ten to the fourth" = $10 \times 10 \times 10 \times 10 = 10{,}000$
• $3^1$ = "three to the power of one" = $3$
• $7^5$ = "seven to the fifth" = $7 \times 7 \times 7 \times 7 \times 7 = 16{,}807$

The second-power case ($a^2$) is so common it has its own name: squared. The third-power case ($a^3$) is called cubed. From $4^4$ onwards, just say "to the power of n".

Common powers to memorise

These come up so often in WAEC that knowing them by heart saves time.

Powers of $2$:

Powers of $3$:

Powers of $5$:

Powers of $10$:

Computing simple powers

For most WAEC questions, you compute powers by direct multiplication.

Example. Find $4^3$.

$4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64$.

Example. Find $5^4$.

$5^4 = 5 \times 5 \times 5 \times 5 = 25 \times 25 = 625$.

Example. Find $6^3$.

$6^3 = 6 \times 6 \times 6 = 36 \times 6 = 216$.

Step-by-step for larger powers

For $2^{10}$, multiply step by step:

$2^1 = 2$ $2^2 = 4$ $2^3 = 8$ $2^4 = 16$ $2^5 = 32$ $2^6 = 64$ $2^7 = 128$ $2^8 = 256$ $2^9 = 512$ $2^{10} = 1024$

For each step, just multiply the previous result by $2$.

Negative bases — careful with signs

When the base is negative, the result depends on whether the exponent is even or odd.

Even exponent: result is positive.

• $(-2)^2 = (-2) \times (-2) = 4$ ✓
• $(-3)^4 = (-3)^2 \times (-3)^2 = 9 \times 9 = 81$ ✓
• $(-5)^2 = 25$ ✓

Odd exponent: result is negative.

• $(-2)^3 = (-2) \times (-2) \times (-2) = 4 \times (-2) = -8$ ✓
• $(-3)^3 = -27$ ✓
• $(-5)^3 = -125$ ✓

The rule: in a product, count the number of negative factors. If even, the product is positive. If odd, the product is negative.

Sign convention — the minus sign before the base

There is a critical difference between $-2^2$ and $(-2)^2$.

• $(-2)^2 = (-2) \times (-2) = 4$ — the parentheses tell you to square the negative number first.
• $-2^2 = -(2^2) = -(4) = -4$ — without parentheses, the minus is outside the exponent. You compute the power first, then apply the negative.

The convention: the exponent applies only to the immediately preceding number or grouping. A minus sign without parentheses is a separate operation applied after the power.

• $-3^2 = -(3 \times 3) = -9$
• $(-3)^2 = (-3) \times (-3) = 9$
• $-2^4 = -(16) = -16$
• $(-2)^4 = 16$

This is one of the most common WAEC traps. Read every minus carefully.

Bases that are fractions

When the base is a fraction, the index applies to the whole fraction.

Examples:

• $\left(\dfrac{1}{2}\right)^3 = \dfrac{1^3}{2^3} = \dfrac{1}{8}$
• $\left(\dfrac{2}{3}\right)^2 = \dfrac{4}{9}$
• $\left(\dfrac{3}{4}\right)^3 = \dfrac{27}{64}$
• $\left(\dfrac{5}{2}\right)^2 = \dfrac{25}{4}$

Notice how the value gets smaller when the fraction is less than $1$, and larger when greater than $1$.

Bases of $0$ and $1$

Two special bases.

Base of $0$

$0^n = 0$ for any positive $n$.

• $0^1 = 0$
• $0^2 = 0$
• $0^5 = 0$

Note: $0^0$ is a special case — sometimes defined as $1$, sometimes undefined. WAEC does not typically test this corner case.

Base of $1$

$1^n = 1$ for any $n$.

• $1^1 = 1$
• $1^2 = 1$
• $1^{100} = 1$
• $1^{-5} = 1$

If you ever see $1$ to a power, the answer is $1$ — no matter the exponent.

Why indices matter

Indices appear everywhere:

Scientific notation

• $34{,}000 = 3.4 \times 10^4$
• $0.0056 = 5.6 \times 10^{-3}$

Areas and volumes

• Area of a square with side $s$: $s^2$
• Volume of a cube with side $s$: $s^3$
• Volume of a sphere: $\dfrac{4}{3}\pi r^3$

Growth and decay

• Population after $t$ years: $P_0 \times r^t$ (where $r$ is the growth rate)
• Compound interest: $P(1 + r)^n$

Geometric sequences

• $1, 2, 4, 8, 16, 32, \ldots$ — each term is the previous one times $2$. The $n$-th term is $2^{n-1}$.
• $1, 3, 9, 27, 81, \ldots$ — $n$-th term is $3^{n-1}$.

Later topics (sequences, financial arithmetic, variation) all rely on a solid understanding of indices.

Comparing values raised to powers

Which is bigger, $3^4$ or $4^3$? Don't guess — compute.

$3^4 = 81$. $4^3 = 64$. So $3^4 > 4^3$.

Which is bigger, $2^{10}$ or $10^2$? $2^{10} = 1024$, $10^2 = 100$. So $2^{10} \gg 10^2$.

Which is bigger, $\dfrac{1}{2^3}$ or $\dfrac{1}{3^2}$? Compute: $\dfrac{1}{8}$ vs $\dfrac{1}{9}$. Since $\dfrac{1}{8} > \dfrac{1}{9}$, the first is bigger.

WASSCE examination focus

Practice problems

1. Compute $4^3$.

2. Compute $7^2$.

3. Compute $2^8$.

4. Compute $(-3)^4$.

5. Compute $-3^4$.

6. Compute $\left(\dfrac{1}{4}\right)^2$.

7. Compute $\left(\dfrac{2}{5}\right)^3$.

8. Find the value of $2^5 - 3^2$.

9. Find the value of $5^3 + 4^3$.

10. Find $(-1)^{15} + (-1)^{20}$.

11. A square has side $12$ cm. What is its area?

12. A cube has side $5$ cm. What is its volume?

Worked solutions

1. $4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64$.

2. $7^2 = 49$.

3. $2^8 = 256$.

4. $(-3)^4 = 81$ (even exponent → positive).

5. $-3^4 = -(81) = -81$ (minus outside, applied after the power).

6. $\left(\dfrac{1}{4}\right)^2 = \dfrac{1}{16}$.

7. $\left(\dfrac{2}{5}\right)^3 = \dfrac{8}{125}$.

8. $2^5 - 3^2 = 32 - 9 = 23$.

9. $5^3 + 4^3 = 125 + 64 = 189$.

10. $(-1)^{15} = -1$ (odd). $(-1)^{20} = 1$ (even). Sum: $-1 + 1 = 0$.

11. Area = $12^2 = 144$ cm².

12. Volume = $5^3 = 125$ cm³.

What's next

In the next lesson we cover the laws of indices — the rules that let you combine, simplify, and manipulate expressions with powers. $a^m \times a^n = a^{m+n}$. $\dfrac{a^m}{a^n} = a^{m-n}$. $(a^m)^n = a^{mn}$. $(ab)^n = a^n b^n$. These laws, with zero, negative and fractional indices and indicial equations, underlie nearly every indices question on WAEC.