Comprehensive theory, key formulas, diagrams, and memory aids for Exponents and Powers.
Scientists often deal with very large and very small numbers. The distance between the Sun and the Earth is about 150,000,000,000 metres, and the size of a dust particle is about 0.0000001 metres. Writing such numbers in full is inconvenient and error-prone. Exponents and powers provide an elegant shorthand for representing very large and very small numbers. Instead of writing a number multiplied by itself many times, we write it using a base and an exponent.
In this chapter we will learn what exponents and powers mean, study the laws of exponents that let us multiply, divide and simplify powers, learn how to handle negative exponents and zero exponent, and learn how to express numbers in standard form (scientific notation). Exponents are used in science, engineering, finance and computers, making them one of the most practical tools in mathematics.
An exponent, also called a power or index, tells us how many times a number (the base) is multiplied by itself. In the expression a^n, the number a is called the base and n is called the exponent.
$$a^n = a \times a \times a \times ... \text{ (n times)}$$
For example, 2^5 = 2 x 2 x 2 x 2 x 2 = 32. Here 2 is the base and 5 is the exponent.
For non-zero integers a and b, and positive integers m and n:
$$a^m \times a^n = a^{m + n}$$
Example: 2^3 x 2^4 = 2^(3+4) = 2^7 = 128.
$$a^m \div a^n = a^{m - n} \quad (m > n)$$
Example: 3^5 / 3^2 = 3^(5-2) = 3^3 = 27.
$$(a^m)^n = a^{mn}$$
Example: (2^3)^2 = 2^(3x2) = 2^6 = 64.
$$a^m \times b^m = (ab)^m$$
Example: 4^3 x 5^3 = (4 x 5)^3 = 20^3 = 8000.
$$a^m \div b^m = \left(\frac{a}{b}\right)^m$$
Example: 6^2 / 3^2 = (6/3)^2 = 2^2 = 4.
Any non-zero number raised to the power zero is 1: a^0 = 1.
$$a^{-n} = \frac{1}{a^n}$$
For example, 2^-3 = 1/2^3 = 1/8. A negative exponent means we take the reciprocal of the number with the positive exponent.
$$a^{-m} \times a^{-n} = a^{-(m+n)}$$ $$\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$$
For example, (2/3)^-2 = (3/2)^2 = 9/4.
Any number can be expressed in standard form as: $$k \times 10^n$$ where k is a decimal number between 1 and 10 (i.e. 1 <= k < 10) and n is an integer.
Standard form is also called scientific notation, and it is widely used in physics, chemistry, astronomy and computer science.
| Law | Statement |
|---|---|
| Multiplication | a^m x a^n = a^(m + n) |
| Division | a^m / a^n = a^(m - n) |
| Power of a power | (a^m)^n = a^(mn) |
| Same exponent product | a^m x b^m = (ab)^m |
| Same exponent quotient | a^m / b^m = (a/b)^m |
| Zero exponent | a^0 = 1 |
| Negative exponent | a^(-n) = 1/a^n |
| Power | Value | Power | Value |
|---|---|---|---|
| 2^3 | 8 | 3^3 | 27 |
| 2^4 | 16 | 3^4 | 81 |
| 2^5 | 32 | 5^2 | 25 |
| 2^6 | 64 | 5^3 | 125 |
| 2^7 | 128 | 10^3 | 1000 |
| 2^8 | 256 | 10^4 | 10000 |
graph TD
A["Exponents and Powers"] --> B["a^n: a is base, n is exponent"]
A --> C["Laws of exponents"]
C --> D["a^m x a^n = a^(m+n)"]
C --> E["a^m / a^n = a^(m-n)"]
C --> F["(a^m)^n = a^(mn)"]
C --> G["a^m x b^m = (ab)^m"]
C --> H["a^m / b^m = (a/b)^m"]
A --> I["Special exponents"]
I --> J["a^0 = 1"]
I --> K["a^(-n) = 1/a^n"]
A --> L["Standard form: k x 10^n, 1 <= k < 10"]
Exponents and powers make it easy to work with very large and very small numbers. In this chapter we learnt the meaning of the base and the exponent, and studied the five main laws of exponents that help us multiply, divide and simplify powers. We also learnt about the special rules for zero and negative exponents, understanding that a^0 = 1 and a^(-n) = 1/a^n. Finally, we studied standard form (scientific notation) and used it to express huge and tiny numbers compactly. These concepts are the backbone of science, engineering and computer science, and they lay the foundation for logarithms, growth and decay problems, and all of higher mathematics.