Powers and Roots: The Complete Guide
1. The Basics: Squares and Cubes
A power (or index) tells you how many times to multiply a number by itself.
Squaring: Multiplying a number by itself once (\(x^2\)).
\(5^2 = 5 \times 5 = 25\)
Cubing: Multiplying a number by itself twice (\(x^3\)).
\(2^3 = 2 \times 2 \times 2 = 8\)
Roots are the opposite of powers. They ask: "What number was multiplied to get this result?"
Square Root: \(\sqrt{49} = 7\) (because \(7 \times 7 = 49\))
Cube Root: \(\sqrt[3]{64} = 4\) (because \(4 \times 4 \times 4 = 64\))
2. Special Powers: Zero and Negative
- Power of Zero: Anything (except zero) to the power of 0 is 1.
\(10^0 = 1\), \(x^0 = 1\), \(527^0 = 1\)
- Negative Powers: A negative index means "take the reciprocal" (turn it into a fraction).
\(x^{-a} = \frac{1}{x^a}\)
\(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\)
3. Fractional Powers (Roots in Disguise)
When the index is a fraction, the bottom number (denominator) is the root, and the top number (numerator) is the power.
$$x^{\frac{1}{n}} = \sqrt[n]{x}$$
\(16^{\frac{1}{2}} = \sqrt{16} = 4\)
\(27^{\frac{1}{3}} = \sqrt[3]{27} = 3\)
If the fraction has a top number other than 1:
\(8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4\)
4. The Laws of Indices
When working with the same base (the big number), we can use these rules to simplify:
| Rule Name |
Formula |
Example |
| Multiplication Rule |
\(a^m \times a^n = a^{m+n}\) |
\(2^6 \times 2^4 = 2^{10}\) |
| Division Rule |
\(a^m \div a^n = a^{m-n}\) |
\(5^7 \div 5^3 = 5^4\) |
| Power of a Power |
\((a^m)^n = a^{m \times n}\) |
\((3^2)^4 = 3^8\) |
5. Combining Integers and Fractions
The rules still apply even if the powers are fractions! Just add or subtract the fractions as normal.
Example 1 (Multiplication):
\(3^7 \times 3^{\frac{1}{2}} = 3^{7 + \frac{1}{2}} = 3^{7.5}\) or \(3^{\frac{15}{2}}\)
Example 2 (Division):
\(x^5 \div x^{\frac{3}{2}} = x^{5 - 1.5} = x^{3.5}\) or \(x^{\frac{7}{2}}\)
Top Exam Tip: You can only combine these if the base is the same.
\(2^3 \times 5^2\) cannot be simplified using index laws!