Powers & Roots

A power is repeated multiplication

Multiplication is repeated addition: $4 \times 3 = 3 + 3 + 3 + 3$ . A power is the next step up, repeated multiplication:

\textcolor{#3b82f6}{2}^{\textcolor{#f43f5e}{5}} = 2 \times 2 \times 2 \times 2 \times 2 = 32

The big number at the bottom is the base; the small raised number is the exponent, and it counts how many copies of the base get multiplied. Two exponents have their own names, borrowed from geometry: a square of side 5 has area $5^2$ (“5 squared”), and a cube of side 5 has volume $5^3$ (“5 cubed”).

Why size changes everything

Powers show up whenever you scale something. Make every length of an object $k$ times bigger, and its area grows by $k^2$ while its volume grows by $k^3$ . Slide the scale factor and count the little squares and cubes:

Scale it up: the square–cube law

Interactive

side2

area4

volume8

area ÷ volume0.5

Scale factor k (the side length)2

The math, with your numbers

2^2 = 2 \times 2 = 4, \qquad 2^3 = 2 \times 2 \times 2 = 8

Area ÷ volume = k² ÷ k³ = 1/k: the bigger the thing, the less surface it has for each unit of bulk.

Try this

→Set k = 2. Count the unit squares and the unit cubes. Do they match 2² and 2³?
→At what k is the volume 27 times the original?
→Watch 'area ÷ volume' as k grows. Why do big animals have trouble keeping cool?

The same scale factor k, applied to a square and to a cube.

This is the square–cube law, and it's why there are no ants the size of horses. An animal's weight depends on its volume, but the strength of its legs depends on their cross-section, an area. Scale an ant up 100 times: its weight grows by $100^3 = 1\,000\,000$ , its leg strength by only $100^2 = 10\,000$ . Each leg would carry 100 times more than it could hold.

Make a prediction first

You double the size of a cube-shaped ice block. How much longer does it take to melt, roughly?

Think about it, then click to reveal the answer.

It has $2^3 = 8$ times as much ice to melt, but heat only gets in through its surface, which is just $2^2 = 4$ times bigger. So it takes about $8 / 4 = 2$ times as long. Big things cool down and warm up slowly, which is exactly the “area ÷ volume” reading in the simulation.

Three rules that do all the work

Since an exponent just counts copies, the rules for combining powers follow from counting. Write them out the long way once and you'll never have to memorise them:

$a^m \times a^n = a^{m+n}$	$2^3 \times 2^2 = (2 \cdot 2 \cdot 2)(2 \cdot 2) = 2^5$	Multiplying: add the exponents.
$\frac{a^m}{a^n} = a^{m-n}$	$\frac{2^5}{2^2} = 2^3$	Dividing: subtract them (two pairs cancel).
$(a^m)^n = a^{m \times n}$	$(2^3)^2 = 2^3 \times 2^3 = 2^6$	A power of a power: multiply them.

The dividing rule explains two strange-looking facts. Divide $2^3$ by itself: the answer must be 1, and the rule gives $2^{3-3} = 2^0$ . So $a^0 = 1$ . Divide $2^2$ by $2^5$ : you get $\frac{1}{2^3}$ , and the rule says $2^{-3}$ . A negative exponent means “one over”:

a^0 = 1, \qquad a^{-n} = \frac{1}{a^n}, \qquad 10^{-3} = \frac{1}{1000} = 0.001

Roots undo powers

A square root asks the question backwards: what number, squared, gives this? $\sqrt{49} = 7$ because $7^2 = 49$ . A cube root undoes cubing: $\sqrt[3]{8} = 2$ because $2^3 = 8$ . In the simulation, the side length is the square root of the area and the cube root of the volume.

Roots are powers too. Since $(a^{1/2})^2 = a^{1/2 \times 2} = a^1$ , the half power must be the square root:

\sqrt{a} = a^{1/2}, \qquad \sqrt[3]{a} = a^{1/3}

Speed after falling a height h

\textcolor{#f43f5e}{v} = \sqrt{2\,g\,\textcolor{#3b82f6}{h}}

Say it out loud: “v equals the square root of two times g times h.”

v

Speed at the bottom· measured in m/s

How fast a dropped object is going when it has fallen h metres (ignoring air).

g

Gravity· measured in 9.8 m/s²

How much faster a falling object goes each second near Earth.

h

Height fallen· measured in m

How far it dropped.

Worked example: a stone dropped from 20 m

Step 1 of 4

1
Put the numbers under the root.
$v = \sqrt{2 \times 9.8 \times 20}$

Wrap-up

Key ideas to remember

A power is repeated multiplication: the exponent counts the copies of the base.
Scale lengths by k and areas grow by k², volumes by k³ (the square–cube law).
Multiply powers: add exponents. Divide: subtract. Power of a power: multiply. $a^0 = 1$ , $a^{-n} = 1/a^n$ .
Roots undo powers, and they are powers too: $\sqrt{a} = a^{1/2}$ .

Check your understanding

Question 1 of 4

What is

3^4

Read the full chapterExponents and Scientific NotationSection 1.2 of Algebra and Trigonometry by OpenStax, free in our library.

Sources & further reading

Everything in this lesson agrees with these references. They're all free to read.

A power is repeated multiplication

Multiplication is repeated addition: $4 \times 3 = 3 + 3 + 3 + 3$ . A power is the next step up, repeated multiplication:

\textcolor{#3b82f6}{2}^{\textcolor{#f43f5e}{5}} = 2 \times 2 \times 2 \times 2 \times 2 = 32

Why size changes everything

Scale it up: the square–cube law

Interactive

side2

area4

volume8

area ÷ volume0.5

Scale factor k (the side length)2

The math, with your numbers

2^2 = 2 \times 2 = 4, \qquad 2^3 = 2 \times 2 \times 2 = 8

Area ÷ volume = k² ÷ k³ = 1/k: the bigger the thing, the less surface it has for each unit of bulk.

Try this

→Set k = 2. Count the unit squares and the unit cubes. Do they match 2² and 2³?
→At what k is the volume 27 times the original?
→Watch 'area ÷ volume' as k grows. Why do big animals have trouble keeping cool?

The same scale factor k, applied to a square and to a cube.

Make a prediction first

You double the size of a cube-shaped ice block. How much longer does it take to melt, roughly?

Think about it, then click to reveal the answer.

Three rules that do all the work

Since an exponent just counts copies, the rules for combining powers follow from counting. Write them out the long way once and you'll never have to memorise them:

$a^m \times a^n = a^{m+n}$	$2^3 \times 2^2 = (2 \cdot 2 \cdot 2)(2 \cdot 2) = 2^5$	Multiplying: add the exponents.
$\frac{a^m}{a^n} = a^{m-n}$	$\frac{2^5}{2^2} = 2^3$	Dividing: subtract them (two pairs cancel).
$(a^m)^n = a^{m \times n}$	$(2^3)^2 = 2^3 \times 2^3 = 2^6$	A power of a power: multiply them.

a^0 = 1, \qquad a^{-n} = \frac{1}{a^n}, \qquad 10^{-3} = \frac{1}{1000} = 0.001

Roots undo powers

Roots are powers too. Since $(a^{1/2})^2 = a^{1/2 \times 2} = a^1$ , the half power must be the square root:

\sqrt{a} = a^{1/2}, \qquad \sqrt[3]{a} = a^{1/3}

Speed after falling a height h

\textcolor{#f43f5e}{v} = \sqrt{2\,g\,\textcolor{#3b82f6}{h}}

Say it out loud: “v equals the square root of two times g times h.”

v

Speed at the bottom· measured in m/s

How fast a dropped object is going when it has fallen h metres (ignoring air).

g

Gravity· measured in 9.8 m/s²

How much faster a falling object goes each second near Earth.

h

Height fallen· measured in m

How far it dropped.

Worked example: a stone dropped from 20 m

Step 1 of 4

1
Put the numbers under the root.
$v = \sqrt{2 \times 9.8 \times 20}$

Wrap-up

Key ideas to remember

A power is repeated multiplication: the exponent counts the copies of the base.
Scale lengths by k and areas grow by k², volumes by k³ (the square–cube law).
Multiply powers: add exponents. Divide: subtract. Power of a power: multiply. $a^0 = 1$ , $a^{-n} = 1/a^n$ .
Roots undo powers, and they are powers too: $\sqrt{a} = a^{1/2}$ .

Check your understanding

Question 1 of 4

What is

3^4

Read the full chapterExponents and Scientific NotationSection 1.2 of Algebra and Trigonometry by OpenStax, free in our library.

Sources & further reading

Everything in this lesson agrees with these references. They're all free to read.

Powers & Roots

A power is repeated multiplication

Why size changes everything

Three rules that do all the work

Roots undo powers

Worked example: a stone dropped from 20 m

Wrap-up

Check your understanding

Where physics uses this

Powers & Roots

A power is repeated multiplication

Why size changes everything

Three rules that do all the work

Roots undo powers

Worked example: a stone dropped from 20 m

Wrap-up

Check your understanding

Where physics uses this