Powers and roots: the rules

Fold a sheet of paper in half. Now it has 2 layers. Fold it again and it has 4 layers. After three folds there are 8. Each fold doubles the number of layers.

After six folds you have 2 · 2 · 2 · 2 · 2 · 2 = 64 layers. A row of the same factor like this is shorter as a power: 2⁶ = 64. In this post you learn to work with powers. At the end you simplify roots such as √72.

The same factor again and again

A power is a multiplication with the same factor every time.

5³ = 5 · 5 · 5 = 125

The number 5 is the base. The small number 3 at the top right is the exponent. The exponent says how many factors there are.

Two powers have their own name. You say "a squared" for a². It is the area of a square with side a. You say "a cubed" for a³. It is the volume of a cube with edge a.

A square and a cube

a = 3

Square

Cube

Square

3² = 3 · 3 = 9

Cube

3³ = 3 · 3 · 3 = 27

Show the numbers

a	a²	a³
1	1	1
2	4	8
3	9	27
4	16	64
5	25	125
6	36	216

With a = 3 you count 9 tiles and 27 blocks. With a = 4 it is 16 tiles and 64 blocks. The cube grows much faster than the square.

The four rules

There are four rules for powers. You do not need to learn them by heart. Write each power out as separate factors and count them. The figure below does the same.

Count the factors

MultiplyDividePower of a powerPower of a product

m = 3n = 2

a³ · a² = a³⁺² = a⁵

• from the first power
• from the second power

aaa·aa

Rule

a^m · aⁿ = a^m+n

Exponent

3 + 2 = 5

Multiplying powers

Write out a³ · a². You get (a · a · a) · (a · a). That is 5 factors a.

a^m · aⁿ = a^m+n

When you multiply, you add the exponents. This only works when the bases are the same.

Dividing powers

a⁵ ÷ a² is a fraction. There are 5 factors a above the bar and 2 below it. Two pairs cancel. 3 factors are left.

a^m ÷ aⁿ = a^m−n

When you divide, you subtract the exponents.

A power of a power

(a²)³ means a² · a² · a². That is 3 groups of 2 factors. Together that makes 6 factors.

(a^m)ⁿ = a^m·n

For a power of a power, you multiply the exponents.

A power of a product

(ab)³ means ab · ab · ab. Put the a's together and the b's together. Then you have a · a · a · b · b · b = a³b³.

(ab)ⁿ = aⁿbⁿ

The exponent belongs to every factor inside the brackets. That includes a number:

(3a²)² = 3² · (a²)² = 9a⁴

Combining the rules

Often you need more than one rule. Then work step by step. A single a is a¹.

(a²)³ · a = a⁶ · a¹ = a⁷

(2x³)² · x ÷ x⁴ = 4x⁶ · x ÷ x⁴ = 4x⁷ ÷ x⁴ = 4x³

You can check a rule with numbers. 2³ · 2⁴ = 8 · 16 = 128. And 2⁷ is also 128.

The exponent 0 and negative exponents

Look at this list. Each step to the left divides by 2.

2³ = 8
2² = 4
2¹ = 2

Take one more step to the left. Then you get 2⁰ = 2 ÷ 2 = 1. One more step gives 2⁻¹ = 1 ÷ 2 = 12. The list carries on below 0.

Halving every step

n = 3

← divide by 2times 2 →

2³

2 · 2 · 2 = 8

Show the numbers

n	2 to the power n	As a decimal
−3	1/8	0.125
−2	1/4	0.25
−1	1/2	0.5
0	1	1
1	2	2
2	4	4
3	8	8
4	16	16
5	32	32
6	64	64

The division rule says the same thing. a³ ÷ a³ = a³⁻³ = a⁰. A number divided by itself is 1. So:

a⁰ = 1

This holds for every base except 0.

Now take a² ÷ a⁵. There are 2 factors above the bar and 5 below it. After cancelling, 3 are left below the bar. What remains is 1a³. The division rule gives a²⁻⁵ = a⁻³. So the two are the same.

a⁻ⁿ = 1aⁿ

Two examples:

5⁻² = 15² = 125 = 0.04
10⁻³ = 110³ = 11000 = 0.001

Minus signs and brackets

With a minus sign in front of a power, brackets make the difference.

(−3)² = (−3) · (−3) = 9
−3² = −(3 · 3) = −9

In (−3)² the minus belongs to the base. In −3² it does not. You work out 3² = 9 first. Then you put the minus in front.

Minus times minus is plus. So an even exponent always gives a positive number. With an odd exponent a minus is left: (−2)³ = −8.

(a + b)² is not a² + b²

The rule (ab)ⁿ = aⁿbⁿ holds for a product. It does not hold for a sum. Take a = 3 and b = 2.

(3 + 2)² = 5² = 25
3² + 2² = 9 + 4 = 13

12 is missing. The figure shows where it is.

The square of a + b

a = 3b = 2

• a²
• b²
• ab, twice

(a + b)²

(3 + 2)² = 5² = 25

a² + b²

3² + 2² = 9 + 4 = 13

What a² + b² misses

2 · 3 · 2 = 12

Show the numbers

Part	Area
a²	9
ab	6
ab	6
b²	4
Together: (a + b)²	25

The square with side a + b has four parts. There is a square a² and a square b². Next to them lie two rectangles of a by b. Those two rectangles are what a² + b² leaves out. With a = 3 and b = 2 that is 2 · 3 · 2 = 12.

(a + b)² = a² + 2ab + b²

Simplifying roots

The square root undoes the square. √49 is the positive number whose square is 49. That is 7, because 7² = 49. So a square of 49 tiles has side 7.

You may split the root of a product.

√(ab) = √a · √b

Check it with numbers. √(4 · 9) = √36 = 6. And √4 · √9 = 2 · 3 = 6.

This rule makes a root simpler. Find the largest square that goes into the number. 36 goes into 72, because 72 = 36 · 2.

√72 = √36 · √2 = 6√2

6√2 is the exact value. As a decimal it is about 8.49.

Tiles in squares

Number: 72

Largest square

36 = 6²

Split

72 = 36 · 2

Root

√72 = √36 · √2 = 6√2

About

≈ 8.49

Show the numbers

Number	Square · rest	Root	About
8	4 · 2	2√2	2.83
12	4 · 3	2√3	3.46
18	9 · 2	3√2	4.24
20	4 · 5	2√5	4.47
24	4 · 6	2√6	4.90
27	9 · 3	3√3	5.20
32	16 · 2	4√2	5.66
36	36 · 1	6	6
45	9 · 5	3√5	6.71
48	16 · 3	4√3	6.93
50	25 · 2	5√2	7.07
72	36 · 2	6√2	8.49
98	49 · 2	7√2	9.90

Lay 72 tiles in equal squares that are as big as they can be. You get two squares of 6 by 6. The side 6 goes in front of the root. The number of squares, 2, stays under the root.

The rules in one list

• a^m · aⁿ = a^m+n
• a^m ÷ aⁿ = a^m−n
• (a^m)ⁿ = a^m·n
• (ab)ⁿ = aⁿbⁿ
• a⁰ = 1 and a⁻ⁿ = 1aⁿ
• (a + b)² = a² + 2ab + b²
• √(ab) = √a · √b