Fractional & negative indices

Description

Fractional indices combine powers and roots. The denominator of the fraction is the root; the numerator is the power.

$$a^{\frac{1}{n}} = \sqrt[n]{a}$$ $$a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$$

Examples: $$8^{\frac{1}{3}} = \sqrt[3]{8} = 2$$ $$27^{\frac{2}{3}} = \left(\sqrt[3]{27}\right)^2 = 3^2 = 9$$ $$16^{\frac{3}{4}} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8$$

Negative fractional indices: apply the negative index rule first. $$8^{-\frac{1}{3}} = \frac{1}{8^{\frac{1}{3}}} = \frac{1}{2}$$

Method: always take the root first (easier numbers), then raise to the power.

Common error: taking the power before the root — this makes the arithmetic much harder and risks errors with large numbers. Root first, power second.

Links

Indices Basic indices Surds

Use the $x^{\square}$ key with a fraction in the index: enter $27$, press $x^{\square}$, enter $(2\div 3)$ using the fraction key.