Exponents: a quick recap
In aⁿ, the number a is the base and n is the exponent (or power). For a positive integer n, aⁿ means a multiplied by itself n times:
2⁵ = 2 × 2 × 2 × 2 × 2 = 32
Negative exponents
Look at the pattern: 10³ = 1000, 10² = 100, 10¹ = 10, 10⁰ = 1. Each time the exponent drops by 1, the value is divided by 10. Continuing the pattern:
10⁻¹ = 1/10, 10⁻² = 1/100, and in general a⁻ⁿ = 1/aⁿ (for a ≠ 0)
A negative exponent means "take the reciprocal". So 2⁻³ = 1/2³ = 1/8, and 1/5⁻² = 5² = 25.
Laws of exponents
For non-zero numbers a and b, and integers m and n:
Law | Example |
|---|---|
aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ |
aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5² = 5⁴ |
(aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ |
aᵐ × bᵐ = (ab)ᵐ | 2⁴ × 5⁴ = 10⁴ |
aᵐ ÷ bᵐ = (a/b)ᵐ | 6³ ÷ 2³ = 3³ |
a⁰ = 1 | 7⁰ = 1, (−3)⁰ = 1 |
The first two laws only work when the bases are the same. You cannot simplify 2³ × 3⁴ by adding the exponents.
Example. Simplify (2⁻¹ + 3⁻¹)⁻¹.
= (1/2 + 1/3)⁻¹ = (5/6)⁻¹ = 6/5.
Example. Find m if 5ᵐ ÷ 5⁻³ = 5⁵.
5ᵐ⁻⁽⁻³⁾ = 5ᵐ⁺³ = 5⁵, so m + 3 = 5 and m = 2.
Example. Simplify (−3)⁴ × (5/3)⁴.
Using aᵐ × bᵐ = (ab)ᵐ: (−3 × 5/3)⁴ = (−5)⁴ = 625.
Standard form
Scientists write very large and very small numbers in standard form (scientific notation):
k × 10ⁿ, where 1 ≤ k < 10 and n is an integer
Large numbers
Move the decimal point to the left until one non-zero digit remains before it. The number of places moved is n.
The speed of light ≈ 300 000 000 m/s = 3 × 10⁸ m/s
The distance from the Earth to the Sun ≈ 149 600 000 000 m = 1.496 × 10¹¹ m
Small numbers
Move the decimal point to the right. The exponent is negative.
The thickness of a human hair ≈ 0.00005 m = 5 × 10⁻⁵ m
The size of a red blood cell ≈ 0.000007 m = 7 × 10⁻⁶ m
Comparing numbers in standard form
Compare the powers of 10 first. 3 × 10⁸ is much bigger than 9 × 10⁶, because 10⁸ is 100 times 10⁶.
Example. The mass of the Earth is about 5.97 × 10²⁴ kg and the mass of the Moon is about 7.35 × 10²² kg. Their total mass is
5.97 × 10²⁴ + 0.0735 × 10²⁴ = 6.0435 × 10²⁴ kg.
To add numbers in standard form, first rewrite them with the same power of 10.
Quick summary
a⁻ⁿ = 1/aⁿ, and a⁰ = 1.
With the same base: multiply means add exponents, divide means subtract, power of a power means multiply.
Standard form is k × 10ⁿ with 1 ≤ k < 10.