Powers and Roots
Powers describe repeated multiplication. Roots undo powers. We will begin with small numbers and simple shapes, then use those ideas to understand index laws.
Foundation & Higher: work through the integer powers and roots first. A short, separate Higher extension introduces fractional indices at the end of the worked examples.
Start with repeated multiplication
A power is a short way to write repeated multiplication.
23 = 2 × 2 × 2 = 8
In 23, 2 is the base: the number being multiplied. 3 is the index: the raised number. It is also called the exponent or power.
For a positive whole-number index, the index tells you how many times to use the base as a factor. Here, a factor means one of the numbers in the multiplication.
23 does not mean 2 × 3. Three factors of 2 give 8; 2 × 3 gives 6.
The symbol × means multiply. Later, we will extend this idea to zero and negative indices.
Example Question:
A square has sides of 5 cm. What is its area?
The unit cm means centimetres. A square’s area is side × side, so this is a question about a power. Example 2 shows why.
Exam Tip
Read the base, index and operation separately. Brackets show whether a minus sign is part of the base.
“Work out” asks for a numerical value. “Write as a single power” asks you to keep an answer such as 25.
Expand a small example if a rule slips your mind. You can often rebuild the rule from its meaning.
Method — Powers and roots
To work out a positive whole-number power: write the base as a repeated multiplication, then calculate one step at a time.
Squared means power 2. For example, 42 = 4 × 4 = 16.
Cubed means power 3. For example, 33 = 3 × 3 × 3 = 27.
A root undoes a power. The square-root symbol √ asks which non-negative number, multiplied by itself, gives the value. Non-negative means zero or positive.
√25 = 5 because 5 × 5 = 25.
The cube-root symbol ∛ asks which number, used as a factor three times, gives the value.
∛27 = 3 because 3 × 3 × 3 = 27.
Use the examples below before trying the index laws. Those laws are shortcuts for calculations with powers.
Worked Examples
Example 1 — Read and calculate a power
Identify the base and index in 23, then work it out.
The base is 2. The index is 3, so write three factors of 2:
23 = 2 × 2 × 2
2 × 2 = 4
4 × 2 = 8
23 = 8.
Example 2 — A square and its area
A square has sides of 5 cm. Find its area.
Area = side × side
5 × 5 = 25
Area = 25 cm2. The unit cm2 means square centimetres.
The array has five rows of five equal squares. Each small square represents 1 cm2.
These common squares are useful to recognise. Each comes from multiplying the number by itself:
| Power | Square number |
|---|---|
| 12 | 1 |
| 22 | 4 |
| 32 | 9 |
| 42 | 16 |
| 52 | 25 |
| 62 | 36 |
| 72 | 49 |
| 82 | 64 |
| 92 | 81 |
| 102 | 100 |
| 112 | 121 |
| 122 | 144 |
| 132 | 169 |
| 142 | 196 |
| 152 | 225 |
Use the pattern rather than learning an unexplained list. For example, 42 is 16 and 52 is 25: the next square needs 9 more small squares. We return to this in Think Like a Mathematician.
Example 3 — A cube and its volume
A cube has edges of 3 cm. Find its volume.
A cube has equal length, width and height. Its volume is edge × edge × edge.
3 × 3 = 9
9 × 3 = 27
Volume = 27 cm3. The unit cm3 means cubic centimetres.
Think of three layers, each containing nine unit cubes. The diagram shows the layers separately; stack them to make the cube.
| Power | Cube number |
|---|---|
| 13 | 1 |
| 23 | 8 |
| 33 | 27 |
| 43 | 64 |
| 53 | 125 |
| 103 | 1000 |
Example 4 — Work backwards with roots
A square has area 25 cm2. Find its side length.
Ask: which positive number multiplied by itself gives 25?
5 × 5 = 25
√25 = 5
The side length is 5 cm. Taking a square root undoes squaring.
Now a cube has volume 125 cm3. Find its edge length.
5 × 5 × 5 = 125
∛125 = 5
The edge length is 5 cm. Taking a cube root undoes cubing.
A side or edge is a length, so we use a positive answer.
Example 5 — Keep brackets around a negative base
Work out (−4)2.
The brackets make the whole number −4 the base:
(−4)2 = (−4) × (−4) = 16
A negative multiplied by a negative is positive.
Without brackets, −42 means take the negative of 4 squared. Powers are calculated before that minus sign:
−42 = −(4 × 4) = −16
Both 4 and −4 square to 16. However, √16 means 4, the principal square root: the non-negative answer.
If a question asks for all numbers whose square is 16, give 4 and −4. That is a different question from evaluating √16.
Index laws — start with matching bases
These rules let us combine powers without writing every factor. Multiplication and division need the same base.
Example 6 — Multiply powers with the same base
Write 23 × 22 as a single power.
23 × 22 = (2 × 2 × 2) × (2 × 2)
There are five factors of 2 altogether:
23 × 22 = 25
The indices add because we count all the factors: 3 + 2 = 5.
Another example has three factors of 3:
32 × 31 = (3 × 3) × 3 = 33
With letters, the rule is am × an = am+n.
Here, a names the common base. m and n name the indices. The raised m + n means add those indices.
For contrast, 23 × 32 has different bases. Do not use the add-the-indices rule:
23 × 32 = 8 × 9 = 72
Example 7 — Divide powers with the same base
Write 25 ÷ 22 as a single power. The symbol ÷ means divide.
25 ÷ 22 = (2 × 2 × 2 × 2 × 2) ÷ (2 × 2)
Cancel the two shared factors of 2. Three factors remain:
25 ÷ 22 = 23
The indices subtract because two of the five factors have been cancelled: 5 − 2 = 3.
am ÷ an = am−n, with a non-zero base a.
Non-zero means not equal to 0. We cannot divide by zero.
Check with ordinary numbers: 32 ÷ 4 = 8, and 23 = 8.
Example 8 — Raise a power to another power
Write (23)2 as a single power.
The outside index 2 means use two copies of the whole bracket:
(23)2 = 23 × 23
Each copy contains three factors of 2. Two groups of three make six factors:
(23)2 = 26
(am)n = amn.
The raised mn means m × n. For a power of a power, multiply the indices.
Check: 23 = 8, and 82 = 64. Also, 26 = 64.
Example 9 — Why the zero index gives 1
Work out 20.
First divide a number by itself:
23 ÷ 23 = 8 ÷ 8 = 1
The index law gives the same calculation a different form:
23 ÷ 23 = 23−3 = 20
So 20 = 1.
The same reasoning works for every non-zero base: a0 = 1.
This lesson does not need to define 00. Do not use the rule with a base of zero.
Negative indices — continue the pattern
Each step down this table divides the value by 10. This pattern will also help in Standard Form. Continue through power 0:
| Power | Value |
|---|---|
| 103 | 1000 |
| 102 | 100 |
| 101 | 10 |
| 100 | 1 |
| 10−1 | 1/10 = 0.1 |
| 10−2 | 1/100 = 0.01 |
A negative index means take a reciprocal: 1 divided by the corresponding positive power.
a−n = 1/an, for non-zero a and a positive whole number n. The slash / means division; 1/an means 1 divided by an.
The index laws continue to work with integer indices: positive whole numbers, zero and negative whole numbers, wherever the expressions are defined.
Example 10 — Use a negative index
Work out 2−3.
First calculate the matching positive power:
23 = 2 × 2 × 2 = 8
Then take its reciprocal:
2−3 = 1/8 = 0.125
The negative index does not make this answer negative. It makes it one eighth.
We can also see it through division:
20 ÷ 23 = 2−3
1 ÷ 8 = 0.125
Example 11 — Choose a root in a GCSE problem
A square garden has area 81 m2. A fence goes around all four sides. Find the total fence length.
The unit m means metres. First find a side from the area:
√81 = 9
Each side is 9 m. The perimeter means the total distance around the outside:
4 × 9 = 36
The fence length is 36 m.
Check the side length: 92 = 81. The fence answer uses m, while the area uses m2.
Higher extension — fractional indices
A fractional index can represent a root. We will use positive bases here.
a1/2 = √a: an index of one half means square root.
a1/3 = ∛a: an index of one third means cube root.
This fits the power-of-a-power idea. Raising a1/2 to power 2 multiplies the indices to give 1, so squaring brings us back to a.
Example 12 — Higher: take a root, then a power
Work out 161/2 and 271/3.
161/2 = √16 = 4
271/3 = ∛27 = 3
Now try 82/3.
The denominator 3 tells us to take the cube root. The numerator 2 tells us to square that result.
∛8 = 2
22 = 4
82/3 = (∛8)2 = 4
For a positive base, am/n means take the nth root, then raise it to power m. Here m and n are positive whole numbers.
Use the root first when it gives a small whole number. Keep this extension separate until the integer-index methods feel secure.
Index laws at a glance
For these integer-index rules, use a non-zero base a. The letters m and n stand for integer indices.
| Operation | Rule |
|---|---|
| Multiply the same base | am × an = am+n |
| Divide the same base | am ÷ an = am−n |
| Power of a power | (am)n = amn |
| Zero index | a0 = 1 |
| Negative index | a−n = 1/an (n positive) |
Using a calculator
Look for power and root functions such as x2, xy, √ and ∛. Names and key sequences vary by model; some functions are in a menu or use a shift key.
The x and y on these keys are placeholders for the numbers you enter. Use the square function for a power of 2, or the general power function for another index.
For a negative base, enter the brackets around the negative number. Check that the display shows the expression you intend.
Use 52 = 25 and √25 = 5 to check that you understand your calculator’s entry method. It should support the maths you already understand.
Check by reversing the operation
For a positive number greater than 1, squaring makes it larger. A square root of a number greater than 1 is smaller than that number.
√49 = 7 can be checked by calculating 72 = 49. Likewise, ∛27 = 3 is checked by 33 = 27.
A non-zero base to power 0 gives 1. A negative index such as 10−2 gives 0.01, a small positive value.
If both indices are positive, multiplying same-base powers adds them, so the resulting index is larger than either starting index.
For small powers, expand or calculate both sides to check an index-law answer. Check units in area, volume and length questions.
Further roots and mixed calculations — both tiers
The small number on a root sign tells you which power to undo. A fourth root asks which non-negative number raised to power 4 gives the value. For example, 34 = 81, so the fourth root of 81 is 3.
. Similarly, the fifth root of 32 is 2 because 25 = 32.
Worked example — roots and reciprocals in a calculation
A reciprocal is 1 divided by a non-zero number. The reciprocal of 2 is ½, which is also 2−1. Work out powers and roots before multiplication or division.
18 ÷ √9 + 2−1
= 18 ÷ 3 + ½
= 6 + ½
= 6½
Quick practice — find the fourth root of 16, then calculate 12 ÷ √4 + 4⁻¹.
24 = 16, so the fourth root is 2.
√4 = 2 and 4−1 = ¼. Then 12 ÷ 2 + ¼ = 6¼.
Higher — estimate a power
Use a nearby convenient base to estimate an awkward power: 2.03³ ≈ 2³ = 8. This is an estimate, not an exact value; because 2.03 is greater than 2, the actual cube is slightly greater than 8.
Quick practice — Estimate 4.98² using a nearby whole number, and say whether the exact answer is above or below your estimate.
4.98² ≈ 5² = 25. The exact answer is below 25 because 0 < 4.98 < 5.
Higher — estimating a root and combining fractional indices
A root need not be a whole number. Since 7² = 49 and 8² = 64, √50 lies between 7 and 8, close to 7. Trying 7.1 gives 7.1² = 50.41, so √50 ≈ 7.1 is a useful estimate. The symbol ≈ means “approximately equal to”.
For a negative fractional index, work out the positive power and then take its reciprocal. Since the fourth root of 16 is 2, 163/4 = 2³ = 8. Therefore 16−3/4 = ⅛.
Higher quick practice — place the cube root of 70 between consecutive integers, and evaluate 27−2/3.
4³ = 64 and 5³ = 125, so 4 < ∛70 < 5. The cube root is closer to 4 than to 5.
The cube root of 27 is 3. Square it: 3² = 9. Take the reciprocal for the negative index: 27−2/3 = ⅑.
Think Like a Mathematician
Why do square numbers grow by odd numbers?
Look back at the 5 × 5 array. Start with the navy 4 × 4 square, containing 16 small squares.
Add a column of 4 on the right. Then add a full row of 5 at the bottom.
4 + 5 = 9
16 + 9 = 25
The next square needs 9 new squares. To grow a 5 × 5 square into a 6 × 6 square, we would add 5 + 6 = 11.
25 + 11 = 36
Each step adds two consecutive whole numbers, whose total is odd. That explains the odd-number gaps in the square-number table.
Common mistakes to avoid
Multiplying the base by the index. 23 uses three factors of 2; it is not 2 × 3.
Confusing squares with doubles, or cubes with triples. Square means multiply a number by itself. Cube means use it as a factor three times.
Dividing by 2 to find a square root. √25 asks what squares to 25. The answer is 5, not 12.5.
Adding indices when bases differ. The multiplication rule needs the same base on both powers.
Mixing the index laws. Multiply powers: add indices. Divide powers: subtract indices. Power of a power: multiply indices.
Thinking a0 is zero. For a non-zero base, it is 1.
Thinking a negative index makes the answer negative. It tells you to take a reciprocal.
Losing brackets. (−4)2 is 16, but −42 is −16.
Try it yourself
Start with Questions 1–5, then practise the laws. Show your working. Questions 14–15 are the clearly labelled Higher fractional-index extension.
1. In 34, identify the base and index. Then work out the value.
The base is 3 and the index is 4. Use four factors of 3:
34 = 3 × 3 × 3 × 3
3 × 3 = 9
9 × 3 = 27
27 × 3 = 81
34 = 81.
2. A square has side length 6 cm. Find its area.
Area = side × side
62 = 6 × 6 = 36
The area is 36 cm2.
3. A cube has edge length 4 cm. Find its volume.
Volume = edge × edge × edge
4 × 4 = 16
16 × 4 = 64
The volume is 64 cm3.
4. Work out √49 and ∛64.
7 × 7 = 49, so √49 = 7.
4 × 4 × 4 = 64, so ∛64 = 4.
The roots undo the corresponding powers.
5. Work out (−3)2 and −32.
With brackets, −3 is the base:
(−3)2 = (−3) × (−3) = 9
Without brackets, square 3 before applying the minus:
−32 = −(3 × 3) = −9
6. Write 52 × 53 as a single power.
The bases match. Two factors of 5 and three factors of 5 make five factors.
2 + 3 = 5
52 × 53 = 55.
7. Write 36 ÷ 32 as a single power.
The bases match and are non-zero. Cancel two of the six factors of 3.
6 − 2 = 4
36 ÷ 32 = 34.
8. Write (42)3 as a single power.
The outside power means three copies of 42. Each copy has two factors of 4.
2 × 3 = 6
(42)3 = 46.
9. Work out 70. Explain why.
72 ÷ 72 = 49 ÷ 49 = 1
The division law also gives:
72 ÷ 72 = 70
So 70 = 1. The base is non-zero.
10. Write 2−4 as a fraction, and 10−2 as a decimal.
First work out the positive powers:
24 = 16
102 = 100
Now take reciprocals:
2−4 = 1/16
10−2 = 1/100 = 0.01
11. A student says 23 × 32 = 65. Explain the mistake and find the correct value.
The bases differ, so the add-the-indices rule does not apply.
23 = 8
32 = 9
8 × 9 = 72
The correct value is 72.
12. A square garden has area 144 m2. Find its side length and perimeter.
Find the positive square root:
√144 = 12
The side length is 12 m.
A square has four equal sides:
4 × 12 = 48
The perimeter is 48 m. Check: 122 = 144.
13. Write (24 × 23) ÷ 25 as a single power, then find its value.
Combine the multiplication inside the brackets first:
24 × 23 = 27
Then divide the matching bases:
27 ÷ 25 = 22
The single power is 22, and its value is 4.
14. Higher: work out 811/2 and 641/3.
An index of one half means square root:
811/2 = √81 = 9
An index of one third means cube root:
641/3 = ∛64 = 4
Check: 92 = 81 and 43 = 64.
15. Higher: work out 272/3.
The denominator 3 tells you to take the cube root first:
∛27 = 3
The numerator 2 tells you to square that result:
32 = 9
272/3 = 9.
Frequently Asked Questions
Are index, exponent and power different things?
They are commonly used names for the raised number. In 23, the base is 2 and the index is 3.
Does √16 have two answers?
√16 means the principal square root, 4. A question asking for all numbers whose square is 16 has two answers: 4 and −4.
Can a cube root be negative?
Yes. (−2) × (−2) × (−2) = −8, so ∛(−8) = −2. An odd number of negative factors gives a negative product.
Do index laws work for addition?
Not as taught here. For example, 23 + 22 = 8 + 4 = 12. Adding the indices would give 25 = 32, a different value.
Does every square root give a whole number?
No. For example, √20 lies between 4 and 5 because 42 = 16 and 52 = 25. This lesson’s main root calculations use exact whole-number answers.
Is a negative index the same as a fractional index?
No. A negative sign in the index tells you to take a reciprocal. An index of one half or one third tells you to take a root. The Higher extension introduces these fractional indices.
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