C1.1, C1.3; E1.1, E1.3 - Squares, cubes and roots | Cambridge IGCSE Mathematics

Powers count repeated equal factors; roots reverse that process. You will recognise square and cube numbers, recall the key values, and work out other whole-number powers and exact roots. The same methods apply to Core and Extended.

Squares and square roots

A square floor contains $144$ equal square tiles. How many tiles lie along one edge: $12$ or $72$ ? Halving the total does not find an edge: each row has the same number of tiles as there are rows.

For a smaller floor, seven rows of seven tiles contain $7\times7=49$ tiles. The short notation is $7^2$ , read “seven squared”. The raised $2$ counts the two equal factors; it does not mean multiply by $2$ .

[DIAGRAM: asset_name: A square and its root; asset_slug: square_and_root; caption: Seven rows of seven unit squares give 49. The square root recovers the side length, 7.; description: An exact seven-by-seven pale blue square grid with side lengths 7 labelled below and to the left. Text to the right states seven squared equals forty-nine and square root of forty-nine equals seven.]
Seven rows of seven unit squares give 49. The square root recovers the side length, 7.

Square and square root

The square of a number is that number multiplied by itself. A square number is the square of an integer: $0,1,4,9,16,\ldots$ .

For a non-negative number, the principal square root is the non-negative number whose square gives it. The symbol $\sqrt{49}$ means $7$ , not $\pm7$ .

Learn these squares and read the table backwards for their roots. For instance, the entry $11,121$ means both $11^2=121$ and $\sqrt{121}=11$ .

Number	Square	Number	Square	Number	Square
$1$	$1$	$6$	$36$	$11$	$121$
$2$	$4$	$7$	$49$	$12$	$144$
$3$	$9$	$8$	$64$	$13$	$169$
$4$	$16$	$9$	$81$	$14$	$196$
$5$	$25$	$10$	$100$	$15$	$225$

Worked example 1: finding a square

Question. Work out $12^2$ .

Plan. Multiply $12$ by itself, using ten twelves and two twelves.

\begin{aligned} 12^2&=12\times12 && \text{two equal factors}\\ &=120+24 && \text{split the multiplication}\\ &=144 && \text{add the two products} \end{aligned}

Answer. $144$ .

Check. $144\div12=12$ : twelve rows of twelve use all $144$ tiles.

Now reverse the question: you know the total and want the repeated factor.

Worked example 2: undoing a square

Question. Write down $\sqrt{144}$ .

Plan. Find the non-negative number that multiplies by itself to make $144$ .

\begin{aligned} 12\times12&=144 && \text{use the recalled square}\\ \sqrt{144}&=12 && \text{take the non-negative root} \end{aligned}

Answer. $12$ .

Check. $11^2=121$ and $13^2=169$ ; the square $144$ lies between these neighbouring squares, as expected.

A root symbol and a question about all possible numbers have different jobs. In “find all numbers whose square is $49$ ”, both $7$ and $-7$ work: a negative multiplied by a negative is positive. If this is written $x^2=49$ , the letter $x$ simply stands for the unknown number, and the answers are $x=7$ and $x=-7$ . But $\sqrt{49}$ is the single principal value $7$ . The next lesson explains signed products in detail.

Watch out

$12^2=24$ confuses squaring with doubling. Squaring gives $12\times12=144$ . Also, $\sqrt0=0$ ; a negative number has no real square root, because no number on the real number line squares to a negative value.

Cubes and cube roots

A cube made from small unit cubes has equal numbers along its length, width and height. With four along each edge, one layer contains $4\times4=16$ unit cubes, and four layers contain $16\times4=64$ .

Cube and cube root

The cube of a number is the product of three equal factors. For example, $4^3=4\times4\times4$ . A cube number is the cube of an integer.

The cube root, written $\sqrt[3]{\phantom{8}}$ , reverses cubing: $\sqrt[3]{64}=4$ . The small $3$ tells you which root to take; it is not a multiplier.

These cube/root pairs must be recalled. They are not supplied in the List of formulas.

Number	$1$	$2$	$3$	$4$	$5$	$10$
Cube	$1$	$8$	$27$	$64$	$125$	$1000$

Worked example 3: counting three equal factors

Question. Work out $4^3$ .

Plan. Multiply twice to combine the three factors.

\begin{aligned} 4^3&=4\times4\times4 && \text{write three factors}\\ &=16\times4 && \text{count one layer first}\\ &=64 && \text{count four layers} \end{aligned}

Answer. $64$ .

Check. $64\div4=16$ , and $16\div4=4$ : undoing two multiplications leaves the original factor.

Keep the same cube, but recover its edge instead of its total. A root reverses the operation, just as in the square examples.

Worked example 4: recovering a cube's edge

Question. Write down $\sqrt[3]{64}$ .

Plan. Look for three equal factors with product $64$ .

\begin{aligned} 4\times4\times4&=64 && \text{identify the cube}\\ \sqrt[3]{64}&=4 && \text{recover the repeated factor} \end{aligned}

Answer. $4$ .

Check. $3^3=27$ and $5^3=125$ , so $64$ is between the neighbouring cubes.

The same number can be both a square and a cube: $64=8^2=4^3$ . The root sign tells you which factor to recover, so $\sqrt{64}=8$ whereas $\sqrt[3]{64}=4$ .

Watch out

$\sqrt[3]{216}=216\div3$ is wrong. A cube root asks for three equal factors, not three equal shares. Since $6\times6\times6=216$ , the cube root is $6$ .

Other powers and roots

The pattern continues beyond squares and cubes. In $3^4$ , the base is $3$ and the index or power is $4$ : multiply four copies of $3$ . The index counts factors, not multiplication signs.

For any positive whole-number index, write that many equal factors. An index of $1$ means just one factor, so $9^1=9$ . Rules for zero and negative indices come later; repeated factors explain the positive indices used here.

Worked example 5: a fourth power

Question. Work out $3^4$ .

Plan. Write four factors, then multiply them in pairs.

\begin{aligned} 3^4&=3\times3\times3\times3 && \text{four copies of the base}\\ &=9\times9 && \text{pair the factors}\\ &=81 && \text{multiply} \end{aligned}

Answer. $81$ .

Check. Repeated division reverses the calculation: $81\div3=27$ , $27\div3=9$ , $9\div3=3$ .

Changing from a power to its matching root reverses the same four-factor calculation.

Worked example 6: a fourth root

Question. Work out $\sqrt[4]{81}$ .

Plan. Find the non-negative number whose fourth power is $81$ .

\begin{aligned} 3\times3\times3\times3&=81 && \text{test four equal factors}\\ \sqrt[4]{81}&=3 && \text{take the principal root} \end{aligned}

Answer. $3$ .

Check. $2^4=16$ and $4^4=256$ , so a value between $2$ and $4$ is sensible.

Even-numbered roots, such as square and fourth roots, use the non-negative principal value when the number inside is non-negative. An odd-numbered root reverses an odd power; for instance, $2^5=32$ , so $\sqrt[5]{32}=2$ . Not every root is a whole number: $\sqrt2$ lies between $1$ and $2$ , since $1^2<2<2^2$ . Keeping the root sign is an exact answer when an integer root does not exist.

Key idea

A power asks for a product of equal factors. Its matching root asks which factor was repeated. The small number on the root sign must match the power being undone.

Quick Recap

Squaring uses two equal factors; cubing uses three. Higher positive whole-number indices count factors in the same way.
Recall squares from $1$ to $15$ , and cubes of $1,2,3,4,5$ and $10$ ; read those facts backwards for roots.
The square-root symbol gives the non-negative principal value. Finding all numbers with a given positive square also includes the negative number.
Check a root by raising it to the matching power.

The opening floor has $12$ tiles along each edge, because $12^2=144$ .

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C1.1, C1.3; E1.1, E1.3 - Squares, cubes and roots

Squares and square roots

Cubes and cube roots

Other powers and roots

Quick Recap

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