Edexcel Mathematics (4XMAH) -Unit 1 - 2.1 Use of Symbols- Study Notes- New Syllabus

Index Notation (Fractional, Negative and Zero Powers)

Index notation (powers) is a shorter way of writing repeated multiplication.

\( a^3=a\times a\times a \)

Zero Powers

Any non-zero number raised to the power 0 equals 1.

\( a^0=1 \quad (a\neq0) \)

\( 5^0=1,\; 20^0=1 \)

Negative Powers

A negative index means reciprocal.

\( a^{-n}=\dfrac{1}{a^n} \)

\( 2^{-3}=\dfrac{1}{2^3}=\dfrac{1}{8} \)

Fractional Powers

A fractional index represents a root.

\( a^{\frac{1}{2}}=\sqrt{a} \)

\( a^{\frac{1}{3}}=\sqrt[3]{a} \)

More generally:

\( a^{\frac{m}{n}}=\sqrt[n]{a^m} \)

Key Summary

Zero power → 1

Negative power → reciprocal

Fractional power → root