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.
2. Exponents and Powers
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.
Special Cases
a^1 = a (any number raised to the power 1 is the number itself)
a^0 = 1 (any non-zero number raised to the power 0 is 1)
1^n = 1 (1 raised to any power is 1)
3. Laws of Exponents
For non-zero integers a and b, and positive integers m and n:
Law 1: Multiplying Powers with the Same Base
$$a^m \times a^n = a^{m + n}$$
Example: 2^3 x 2^4 = 2^(3+4) = 2^7 = 128.
Law 2: Dividing Powers with the Same Base
$$a^m \div a^n = a^{m - n} \quad (m > n)$$
Example: 3^5 / 3^2 = 3^(5-2) = 3^3 = 27.
Law 3: Power of a Power
$$(a^m)^n = a^{mn}$$
Example: (2^3)^2 = 2^(3x2) = 2^6 = 64.
Law 4: Multiplying Powers with the Same Exponent
$$a^m \times b^m = (ab)^m$$
Example: 4^3 x 5^3 = (4 x 5)^3 = 20^3 = 8000.
Law 5: Dividing Powers with the Same Exponent
$$a^m \div b^m = \left(\frac{a}{b}\right)^m$$
Example: 6^2 / 3^2 = (6/3)^2 = 2^2 = 4.
4. Negative and Zero Exponents
Zero Exponent
Any non-zero number raised to the power zero is 1: a^0 = 1.
Negative Exponent
$$a^{-n} = \frac{1}{a^n}$$
For example, 2^-3 = 1/2^3 = 1/8. A negative exponent means we take the reciprocal of the number with the positive exponent.
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.
Examples
150,000,000,000 = 1.5 x 10^11
0.0000001 = 1 x 10^-7
59,000 = 5.9 x 10^4
0.0025 = 2.5 x 10^-3
Standard form is also called scientific notation, and it is widely used in physics, chemistry, astronomy and computer science.
Quick Revision Tables
Table 1: Laws of Exponents
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
Table 2: Some Powers to Remember
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
Mind Map
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"]
Important Diagrams (SVG)
Diagram 1: Meaning of an Exponent
Diagram 2: Writing Numbers in Standard Form
Common Mistakes
Students add the bases instead of the exponents when multiplying powers with the same base. For example, 2^3 x 2^4 = 2^7, not 2^12 or 4^7.
Students confuse the power of a power rule. (a^m)^n = a^(mn), not a^(m + n).
Students write a^0 = 0, forgetting that any non-zero number raised to the power zero is 1.
While handling negative exponents, students forget to take the reciprocal and may write 2^-3 = -8 instead of 1/8.
In standard form, students allow k to be greater than or equal to 10, such as writing 59,000 as 59 x 10^3 instead of 5.9 x 10^4.
Students make sign errors with the exponent of 10 for small numbers. For 0.0025, the exponent is -3, not +3.
When dividing powers with different bases but the same exponent, students may wrongly combine the bases; the rule is a^m / b^m = (a/b)^m.
Exam Tips
Write down the exponent law you are using before applying it; this helps you avoid mistakes and earns method marks.
Always simplify step by step, combining the exponents carefully with the correct sign.
Memorise the rule a^0 = 1 and practise negative exponents by converting to fractions.
For standard form questions, first fix k between 1 and 10, then decide the exponent of 10 by counting decimal moves.
Check the sign of the exponent of 10: large numbers use positive exponents, small numbers use negative exponents.
Practise converting back and forth between standard form and normal numbers for speed.
Remember that the laws of exponents apply to non-zero bases in the case of negative and zero exponents.
Conclusion
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.