Exponents · RiskAtoms Academy

By the end you will be able to

1Read a power, name its base and exponent, and work out whole-number powers.
2Use the rules: multiply, divide, power of a power, and the zero exponent.
3Work with negative exponents (one over) and fractional exponents (roots).
4Calculate (1 + r)ⁿ and explain why it grows faster and faster.

Why you need it

Multiply by 1.05 once and not much happens. Do it 30 times and you have 4.32 times what you started with. Exponents are the short way to write “do it again, and again”.

L1.3.02 Simple and compound interest
Growth over many periods: (1 + r)ⁿ.
L1.3.03 Present value and discounting
Shrinking a future amount back to today: (1 + r)⁻ⁿ.
L1.3.08 The yield curve: spot and forward rates
Combining rates for different periods with the rules of exponents.
L1.4.03 Pricing a bond
One factor (1 + y)ᵗ for each payment date.
L1.6.02 FX forwards and covered interest parity
Growth factors over periods that may not be whole years.
L1.7.02 Forwards
Carrying a price forward in time with a growth factor.
L1.7.08 Put-call parity
Bringing a future amount back to today with a negative exponent.

The idea

Repeated multiplication

A power is a short way to write the same multiplication over and over. $2^3$ means “start at 1 and multiply by 2, three times”: 1 × 2 × 2 × 2 = 8. Change the base and the exponent and watch the chain:

Base

Exponent3

1× 2× 2× 2=8

2³ means: start at 1 and multiply by 2, 3 times.

Base 1.05 is the growth factor for 5% a step (see M-A01). Notice how each step adds a little more than the one before, because it multiplies a bigger number.

The rules

The notation. Two numbers, two jobs: the base and the exponent.

The rules. Every rule comes from counting copies of the base. You never need to memorise them if you can rebuild them.

Fractional exponents. The same rules tell us what a power of ½ must mean: a root.

Growth: (1 + r)ⁿ. The power you will meet most in finance.

The rules on one card

a^m \times a^n = a^{m+n} \qquad \frac{a^m}{a^n} = a^{m-n} \qquad (a^m)^n = a^{mn}

a^0 = 1 \qquad a^{-n} = \frac{1}{a^n} \qquad a^{\frac{1}{n}} = \sqrt[n]{a}

These work when the base is the same. There is no simple rule for adding powers.

Multiplying against adding, at 5% a step:

Steps n	Multiply: 1.05ⁿ	Add: 1 + 0.05n	Gap
1	1.0500	1.0500	0.0000
2	1.1025	1.1000	0.0025
5	1.2763	1.2500	0.0263
10	1.6289	1.5000	0.1289
20	2.6533	2.0000	0.6533
30	4.3219	2.5000	1.8219

Try it

Move $n$ up and watch the curve bend upwards while the straight line of adding falls behind. Then switch to a negative exponent and watch the same factor shrink things instead.

Try itPower growth

Open the exponents calculator →

Growth per step, r8.0%

Number of steps, n20

Exponent

(1.080)²⁰ =

4.6610

Adding instead: 1 + 20 × r2.6000

First step added0.0800

Last step added0.3453

multiply each step: (1 + r)ᵏadd r each step: 1 + k r

Across: the step k. Up: the growth factor. The purple line marks where the start has doubled.

Worked examples

1. Growth of 5% a step for 10 steps

Step 1 of 3

1
Write it as a power
$(1 + 0.05)^{10} = 1.05^{10}$ . On a calculator: 1.05 ^ 10.

2. Three steps backwards at 4%, starting from 1,000

Step 1 of 3

1
A negative exponent
$1{,}000 \times 1.04^{-3} = \frac{1{,}000}{1.04^3}$

3. Twelve equal monthly steps that make 6% over a year

Step 1 of 3

1
Set up the equation
Twelve equal steps of factor $m$ must make 1.06: $m^{12} = 1.06$ .

Common mistakes

Each of these is tempting. Decide what is wrong before you reveal the fix.

“2³ = 2 × 3 = 6.”
“x² × x³ = x⁶.”
“2⁻³ = −8.”
“5⁰ = 0.”
“(1 + r)² = 1 + r².” Or, on a calculator: 1 + 0.05 ^ 10.

Reaction

Check what stuck

Question 1 · objective 4

Calculate 1.03⁴, to four decimal places.

Question 2 · objective 2

Simplify x³ × x⁴.

Ax¹²Bx⁷C2x⁷Dx⁻¹

Question 3 · objective 3

What is 2⁻³ as a decimal?

Question 4 · objective 3

What is 1.44^(1/2)?

Question 5 · objective 4

r is positive and n is a whole number greater than 1. Which is bigger, (1 + r)ⁿ or 1 + n r?

A(1 + r)ⁿB1 + n rCThey are always equalDIt depends on the base

Glossary

Power: A number multiplied by itself a given number of times, written with a raised exponent: 2³ = 2 × 2 × 2.
Base: The number being multiplied in a power: the 2 in 2³.
Exponent: The raised number in a power, counting how many copies of the base are multiplied: the 3 in 2³. Also called the power or index.
Root: The number that, multiplied by itself n times, gives a. The square root of 9 is 3; the cube root of 8 is 2.
Growth factor: 1 + r: what you multiply by for one step of growth r.
Negative exponent: One over the power: a⁻ⁿ = 1 ÷ aⁿ.