Further Powers and Roots

Work out powers beyond cubes, like 2 to the power 5, and find square, cube and fourth roots on your calculator, then check them by powering back.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 24 real past-paper questions below

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What Further Powers Mean

A power says how many copies of a number you multiply. Squares use 2 copies and cubes use 3. The power can be any whole number.

Formula

a^{n} = a \times a \times a \times \dots \times a

$a$ is the number being multiplied, called the base. $n$ is the small raised number, the power. It says how many copies of $a$ go in the multiplication.

Worked example

1
$2^{5} = 2 \times 2 \times 2 \times 2 \times 2$ — The power is 5, so write down 5 copies of 2 and multiply them.
2
$2 \times 2 = 4$ — Multiply two at a time. Start with the first two 2s.
3
$4 \times 2 = 8$ — Times the third 2. Each new 2 doubles the answer.
4
$8 \times 2 = 16$ — Times the fourth 2. It doubles again.
5
$16 \times 2 = 32$ — Times the fifth and last 2. All 5 copies are used, so $2^{5} = 32$ .

Rule

A root goes backwards. The fifth root of 32 asks: which number, used 5 times, makes 32? It is 2, written $532 = 2$ .

Key idea

Squares, cubes and their roots have their own lessons. This one is about bigger powers, bigger roots, and your calculator.

Work Out a Power

The 1-mark question: “Find the value of $3^{5}$ .” Multiply by 3 again and again, or use the power key on your calculator.

Worked example

1
$3^{5} = 3 \times 3 \times 3 \times 3 \times 3$ — The power 5 means five 3s multiplied together.
2
$3 \times 3 = 9$ — Multiply the first two 3s.
3
$9 \times 3 = 27$ — Times the third 3.
4
$27 \times 3 = 81$ — Times the fourth 3.
5
$81 \times 3 = 243$ — Times the fifth 3. All five 3s are used, so $3^{5} = 243$ .

The 2-mark question puts two powers together. Work out each power on its own first, then multiply.

Worked example

1
$2^{5} \times 3^{2}$ — The question. Powers come before multiplying, so each power is worked out on its own.
2
$2^{5} = 32$ — First power: five 2s multiplied make 32.
3
$3^{2} = 9$ — Second power: two 3s multiplied make 9.
4
$32 \times 9 = 288$ — Now multiply the two values to finish.

Powers added work the same way: each power first, then add.

Worked example

1
$5^{2} + 2^{4}$ — The question. Powers come before adding, so do each power first.
2
$5^{2} = 25$ — Two 5s multiplied: $5 \times 5 = 25$ .
3
$2^{4} = 16$ — Four 2s multiplied: 2, 4, 8, 16.
4
$25 + 16 = 41$ — Last, add the two values.

Key idea

Write each power’s value down before you multiply or add. In a 2-mark question one mark is for those values, so you still score if the last step goes wrong.

Common mistake

Writing $3^{5} = 15$ (that is $3 \times 5$ ), or $2^{2} + 2^{3} = 2^{5}$ . Powers only join up like that when you multiply. Here $4 + 8 = 12$ , not 32.

Powers of Negative Numbers

A minus sign is easy to lose. Put the negative number in brackets and multiply it one copy at a time.

Worked example

1
$(- 2)^{4} = (- 2) \times (- 2) \times (- 2) \times (- 2)$ — The bracket means the whole −2, minus sign included, is used 4 times.
2
$(- 2) \times (- 2) = 4$ — Minus times minus is plus.
3
$4 \times (- 2) = - 8$ — Plus times minus is minus.
4
$(- 8) \times (- 2) = 16$ — Minus times minus is plus again. So $(- 2)^{4} = 16$ .

Worked example

1
$(- 3)^{3} = (- 3) \times (- 3) \times (- 3)$ — Now an odd power: three copies of −3.
2
$(- 3) \times (- 3) = 9$ — The first two minus signs make a plus.
3
$9 \times (- 3) = - 27$ — The third minus sign has no partner, so the answer is negative.

Rule

A negative number to an even power is positive. To an odd power it is negative.

Common mistake

Writing $- 2^{4}$ with no brackets. That means $- (2^{4}) = - 16$ : the power uses only the 2, and the minus goes on after. On a calculator, type the brackets: (−2) then the power key, then 4.

Roots on a Calculator

The most common question: “Find the cube root of 12 167” or “Find $20.25$ .” Estimate, use the root key, then check.

Worked example

1
$4^{2} = 16, 5^{2} = 25$ — Estimate first. 20.25 is between 16 and 25, so its square root is between 4 and 5.
2
$20.25 = 4.5$ — Press the square root key, then 20.25. 4.5 is between 4 and 5, so it fits.
3
$4.5 \times 4.5 = 20.25$ — Check by squaring back. You get the number you started with, so 4.5 is right.

Worked example

1
$2 0^{3} = 8000, 3 0^{3} = 27000$ — Estimate first. 12 167 is between them, so its cube root is between 20 and 30.
2
$312167 = 23$ — Press the cube root key, then 12 167. 23 is between 20 and 30, so it fits.
3
$23 \times 23 = 529$ — Check by cubing back. Multiply two 23s first.
4
$529 \times 23 = 12167$ — Times the third 23. It matches, so 23 is right.

Common mistake

Dividing by 3: 12 167 ÷ 3 is about 4056. A cube root is the number that is multiplied by itself 3 times.

Fourth Roots and Beyond

The fourth root asks: which number, used 4 times, makes this? Use the root key where you type the small number: type 4, then the number.

Worked example

1
$5^{4} = 625, 7^{4} = 2401$ — Estimate first. 1296 is between them, so its fourth root is between 5 and 7.
2
$41296 = 6$ — Root key with the small 4, then 1296. 6 is between 5 and 7, so it fits.
3
$6 \times 6 = 36$ — Check by powering back. Two 6s make 36.
4
$36 \times 36 = 1296$ — Four 6s is two lots of 36 multiplied. It matches, so 6 is right.

Common mistake

Dividing by 4: $416$ is not 4. It is 2, because $2 \times 2 \times 2 \times 2 = 16$ .

Key idea

No root key? A fourth root is a square root twice: $81 = 9$ , then $9 = 3$ . So $481 = 3$ .

Rule

A negative number has no fourth root, because any number to the power 4 is positive. Odd roots are fine: $5 - 32 = - 2$ , because $(- 2)^{5} = - 32$ .

Find the Number Under the Root

The 1-mark question can give you the root and ask for the number inside. To undo a root, use the power that matches it.

Formula

n x = a ⟹ x = a^{n}

$x$ is the number under the root sign, $n$ is the small number on the root (2 if there is none), and $a$ is the root’s value.

Worked example

1
$x = 25$ — No small number on the root, so it is a square root. Undo it by squaring.
2
$x = 25^{2}$ — Square both sides. Squaring undoes the square root and leaves $x$ alone.
3
$x = 25 \times 25 = 625$ — Two 25s multiplied.
4
$625 = 25$ — Check with the root key. It gives back 25, so 625 is right.

Worked example

1
$4 x = 3$ — A fourth root, so undo it with the power 4.
2
$x = 3^{4}$ — Raise both sides to the power 4.
3
$3 \times 3 = 9$ — Four 3s, two at a time. The first two make 9.
4
$9 \times 9 = 81$ — The other two 3s also make 9, so multiply 9 by 9. So $x = 81$ .

Common mistake

Taking the root again ( $25 = 5$ ) or doubling (25 × 2 = 50). To undo a root, you power.

Which Method? Read the Question

Look at the small number: the power, or the number on the root sign.

You see	You do
Find the value of a power	multiply the copies, or power key
Two powers multiplied	work out each one, then multiply
Two powers added	work out each one, then add
A negative number to a power	keep it in brackets: even power +, odd power −
Cube root or fourth root	root key, then power it back
The root is given, find the number	power it back: √x = 25 gives x = 25²
Squares up to 15 squared	Square Numbers
Square root without a calculator	Square Roots
Cubes of 1 to 10	Cube Numbers
Side of a cube from its volume	Cube Roots
Write as a single power	Laws of Indices
A power of 0 or a negative power	Zero and Negative Indices
A power like ½ or ⅓	Fractional Indices

Key idea

Always check a root by powering it back. If you get the number you started with, it is right.

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