POP

G1.9 Powers and Roots

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01 — Explanation

What does a power actually mean?

A power (or index) is shorthand for repeated multiplication of the same number. In $a^n$ , $a$ is the base being multiplied, and $n$ is the power telling you how many times to multiply it by itself.

2^3 = 2 \times 2 \times 2 = 8

Common Mistake

Reading $2^3$ as " $2 \times 3$ ." The power tells you how many times to multiply the base by itself, not what to multiply it by — $2^3$ is three 2's multiplied together, giving $8$ , not $6$ .

02 — Explanation

What is a square root, and how is it the opposite of squaring?

The square root of a number asks "what value, multiplied by itself, gives this number?" It undoes squaring: if $b^2 = a$ , then $\sqrt{a} = b$ .

\sqrt{25} = 5 \quad \text{because} \quad 5^2 = 25 \qquad\qquad \sqrt{16}=4, \quad \sqrt{81}=9

Exam Tip

Memorise the square numbers from $1^2$ to $15^2$ (1, 4, 9, 16, 25, ..., 225) — being able to instantly recognise, say, $169$ as $13^2$ saves significant time on non-calculator papers.

03 — Explanation

What is a cube root, and how does it relate to cubing?

The cube root asks "what value, multiplied by itself three times, gives this number?" It undoes cubing: if $b^3=a$ , then $\sqrt[3]{a}=b$ .

\sqrt[3]{8} = 2 \quad \text{because} \quad 2^3 = 2\times2\times2 = 8 \qquad\qquad \sqrt[3]{27}=3, \quad \sqrt[3]{125}=5

Common Mistake

Confusing $\sqrt[3]{a}$ with $\sqrt{a}$ . Checking your answer by reversing the operation catches this instantly — if you claim $\sqrt[3]{64}=4$ , check: $4^3 = 64$ ✓, but if you'd said $\sqrt{64}$ instead, the check would need $4^2=64$ , which is false (it's actually $8^2=64$ ).

04 — Explanation

How do powers and roots work together in exam questions?

Questions often ask you to evaluate an expression mixing both, or to find a missing power. Work step by step, resolving each power or root as its own calculation before combining them.

Example: Evaluate $\sqrt{36} + 2^3$ .

$\sqrt{36} = 6$ (since $6^2=36$ ).
$2^3 = 8$ .
$6 + 8 = 14$ .

05 — Explanation

What happens with powers of 1, 0, and 10?

A few special cases are worth knowing by sight rather than working out each time: $1^n=1$ for any power $n$ (1 multiplied by itself is always 1), $a^1=a$ (a single "multiplication" does nothing), and $10^n$ just adds $n$ zeros after the 1 (e.g. $10^4=10000$ ).

Common Mistake

Writing $a^0=0$ . Any non-zero number raised to the power $0$ equals $1$ , not $0$ — this is a fixed rule you must memorise rather than derive from repeated multiplication, since "multiplying zero times" isn't something you can picture directly.

06 — Worked example

Worked Exam-Style Question

A square garden has an area of $144 \text{ m}^2$ . (a) Find the length of one side. (b) A cube-shaped water tank has a volume of $125 \text{ m}^3$ . Find the length of one edge.

(a) The area of a square is (side length) $^2$ , so the side length is $\sqrt{144}$ . Since $12^2=144$ , the side length is $12$ m.
(b) The volume of a cube is (edge length) $^3$ , so the edge length is $\sqrt[3]{125}$ . Since $5^3=125$ , the edge length is $5$ m.

Exam Tip

Recognising that "area of a square" needs a square root and "volume of a cube" needs a cube root is a very common Grade 1–2 link between shape formulas and roots — always match the root to the power used to build the formula.