⚡️ Fractional Powers Unleashed! Mastering Rational Indices

Indices don’t stop at whole numbers! Understanding **rational and fractional indices** lets you simplify roots, solve exponential equations and tackle surds with confidence. Mastery of this topic is a high-yield skill for O-Level exams because it threads into algebra, coordinate geometry, and even logarithms.

Essential Rules for Rational & Fractional Indices

For any positive $a$ : $a^{\frac{m}{n}} = \sqrt[n]{a^{\,m}} = \bigl(\sqrt[n]{a}\bigr)^{m}$ .
Negative powers flip the base: $a^{-\frac{m}{n}} = \dfrac{1}{a^{\frac{m}{n}}}$ .
**Add** powers when bases match: $a^{p}\times a^{q}=a^{p+q}$ (works for fractions too).
**Multiply** powers when the same exponent is on different bases: $a^{r}b^{r}=(ab)^{r}$ .

Worked Example: Simplify $27^{2/3}$ and $16^{-3/4}$

Problem: Simplify

27^{\frac{2}{3}}

and

16^{-\frac{3}{4}}

. Step 1 – Rewrite the first expression using roots:

27^{\frac{2}{3}} = \bigl(\sqrt[3]{27}\bigr)^{2}

. Step 2 – Evaluate the cube root:

\sqrt[3]{27}=3

, so the expression becomes

3^{2}=9

. Step 3 – Apply the negative power rule on the second expression:

16^{-\frac{3}{4}} = \dfrac{1}{16^{\frac{3}{4}}}

. Step 4 – Convert to a root:

16^{\frac{3}{4}} = \bigl(\sqrt[4]{16}\bigr)^{3}

. Step 5 – Evaluate the fourth root:

\sqrt[4]{16}=2

, hence

2^{3}=8

. Step 6 – Combine:

16^{-\frac{3}{4}} = \dfrac{1}{8}

. Therefore,

27^{\frac{2}{3}} = 9

and

16^{-\frac{3}{4}} = \dfrac{1}{8}

. Keep practising similar problems to lock in these rules and speed up your exam performance!

Essential Rules for Rational & Fractional Indices

Worked Example: Simplify 272/327^{2/3}272/3 and 16−3/416^{-3/4}16−3/4

Worked Example: Simplify $27^{2/3}$ and $16^{-3/4}$