Exponents, also called powers or indices, are a shorthand way of writing repeated multiplication. Instead of writing 2 x 2 x 2 x 2 x 2, we write 2^5, read as 2 raised to the power 5 or 2 to the power 5. Here 2 is called the base and 5 is called the exponent. The expression 2^5 means that the base 2 is multiplied by itself 5 times. Exponents make very large and very small numbers easy to write and compare. For example, the distance of the Earth from the Sun is about 1.5 x 10^8 km, and the mass of a dust particle is about 1 x 10^-9 grams.
The concept of exponents is essential in science, engineering and economics, where numbers vary from the size of atoms to the size of galaxies. In this chapter we learn the laws of exponents, how to express numbers in exponential form, how to convert large and small numbers into standard form using powers of 10, and how to compare numbers written with exponents. A strong grasp of exponents prepares students for scientific notation, logarithms and growth calculations in higher mathematics.
For any non-zero base a and positive integer n, a^n means a multiplied by itself n times: a^n = a x a x a x ... n times For example, 3^4 = 3 x 3 x 3 x 3 = 81, and 2^3 = 8. The number a^1 = a, and by convention a^0 = 1 for any non-zero a.
Raising a number to the power 2 is called squaring it, and the power 3 is called cubing it. For example, 5^2 = 25 is read as 5 squared, and 4^3 = 64 is read as 4 cubed.
When the base is negative, the sign of the result depends on whether the exponent is even or odd. A negative base raised to an even power gives a positive result, while an odd power gives a negative result: (-2)^4 = 16 and (-2)^3 = -8. Note that (-2)^4 is different from -2^4 = -16, where the negative sign is not included in the base.
When multiplying powers with the same base, add the exponents: a^m x a^n = a^(m+n) For example, 2^3 x 2^4 = 2^7 = 128.
When dividing powers with the same base, subtract the exponents: a^m / a^n = a^(m-n), for a not equal to 0 For example, 5^6 / 5^2 = 5^4 = 625.
When raising a power to another power, multiply the exponents: (a^m)^n = a^(m x n) For example, (3^2)^3 = 3^6 = 729.
The power of a product is the product of the powers: (a x b)^m = a^m x b^m For example, (2 x 3)^2 = 2^2 x 3^2 = 4 x 9 = 36.
The power of a quotient is the quotient of the powers: (a/b)^m = a^m / b^m, for b not equal to 0 For example, (2/3)^3 = 8/27.
For any non-zero a: a^0 = 1. Also, a^(-n) = 1/a^n, the reciprocal of a^n. For example, 7^0 = 1 and 2^(-3) = 1/8.
Any number can be expressed in exponential form by writing its prime factorisation as a product of powers. For example, 8 = 2 x 2 x 2 = 2^3, 81 = 3 x 3 x 3 x 3 = 3^4, and 144 = 2^4 x 3^2. To express a number in exponential form, first factorise it into primes, then group repeated factors as powers.
Very large and very small numbers are written in standard form as a number between 1 and 10 multiplied by a power of 10: Standard form = a x 10^n, where 1 is less than or equal to a, and a is less than 10. For example, the distance from the Earth to the Moon is about 384400000 m = 3.844 x 10^8 m. A small number like 0.0000067 is written as 6.7 x 10^(-6). To convert, move the decimal point so that one non-zero digit lies before it, and count the number of places moved: moving left gives a positive power, moving right gives a negative power.
To compare numbers written in exponential or standard form, first compare the exponents of 10. The number with the larger power of 10 is larger. If the powers are equal, compare the decimal coefficients. For example, 3.2 x 10^9 is greater than 8.7 x 10^8 because the exponent 9 > 8, regardless of the coefficients.
| Law | Statement | Example |
|---|---|---|
| Product | a^m x a^n = a^(m+n) | 2^3 x 2^4 = 2^7 |
| Quotient | a^m / a^n = a^(m-n) | 5^6 / 5^2 = 5^4 |
| Power of a power | (a^m)^n = a^(m x n) | (3^2)^3 = 3^6 |
| Power of a product | (a x b)^m = a^m x b^m | (2 x 3)^2 = 36 |
| Power of a quotient | (a/b)^m = a^m / b^m | (2/3)^3 = 8/27 |
| Zero exponent | a^0 = 1 | 7^0 = 1 |
| Negative exponent | a^(-n) = 1/a^n | 2^(-3) = 1/8 |
| Number | Prime Factorisation | Exponential Form |
|---|---|---|
| 8 | 2 x 2 x 2 | 2^3 |
| 81 | 3 x 3 x 3 x 3 | 3^4 |
| 144 | 2 x 2 x 2 x 2 x 3 x 3 | 2^4 x 3^2 |
| 1000 | 10 x 10 x 10 | 10^3 |
| 0.0005 | 5/10000 | 5 x 10^(-4) |
Exponents and powers give us a compact notation for repeated multiplication and a powerful tool for working with very large and very small numbers. The laws of exponents, which add, subtract and multiply exponents under the right conditions, allow complex expressions to be simplified quickly. Writing numbers in exponential form through prime factorisation and in standard form as a x 10^n connects mathematics with science, where the size of the universe and the size of an atom are both conveniently expressed with powers of 10. Careful attention to the sign conventions and the order of operations prevents common errors. This chapter is essential for scientific notation, logarithms and exponential growth in higher classes.