Laws of indices

An index (or exponent) tells you how many times a base is used as a factor. For example, \(2^4 = 2 \times 2 \times 2 \times 2\). The laws of indices let us simplify expressions efficiently and work with very large or very small quantities.

Meaning of indices

Core laws of indices

Negative indices

A negative index means reciprocal: \[ a^{-n} = \frac{1}{a^n}, \qquad a \neq 0. \]

Example. \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\).

Example. \(x^{-2}y^3 = \frac{y^3}{x^2}\).

Fractional indices

Fractional indices represent roots: \[ a^{1/n} = \sqrt[n]{a}, \qquad a^{m/n} = \sqrt[n]{a^m}. \]

Example. \(27^{1/3} = 3\).

Example. \(16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8\).

Simplification examples

Example 1. Simplify \(a^5 \cdot a^{-2}\).

\(a^{5+(-2)} = a^3\).

Example 2. Simplify \(\dfrac{3x^4y^{-1}}{6x^2}\).

\[ \frac{3x^4y^{-1}}{6x^2} = \frac{1}{2}x^{4-2}y^{-1} = \frac{x^2}{2y}. \]

Example 3. Simplify \((2a^3b^{-2})^2\).

\(2^2 a^{6} b^{-4} = \dfrac{4a^6}{b^4}\).

Scientific notation link

Indices are central to scientific notation: \[ N = k \times 10^n, \qquad 1 \leq k < 10. \] Positive \(n\) gives large numbers, negative \(n\) gives small decimals.

Example. \(0.00045 = 4.5 \times 10^{-4}\).

History

Exponent ideas developed over centuries as mathematicians searched for compact ways to write repeated multiplication. Ancient methods used words or special marks, but a systematic notation appeared gradually in early modern Europe.

In the 17th century, mathematicians such as Descartes helped popularize writing powers with superscripts (like \(x^2\), \(x^3\)). Later work by Wallis and Newton expanded these ideas to fractional and negative exponents. This notation became a foundation for algebra, calculus, and scientific notation.

Derivation

Why \(a^0 = 1\)

Using the quotient law with \(a \neq 0\): \[ \frac{a^m}{a^m} = a^{m-m} = a^0. \] But \(\dfrac{a^m}{a^m} = 1\), so \(a^0 = 1\).

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

Again from the quotient law: \[ \frac{a^0}{a^n} = a^{0-n} = a^{-n}. \] Since \(a^0 = 1\), we get \(a^{-n} = \dfrac{1}{a^n}\).

Why \(a^{1/n} = \sqrt[n]{a}\)

Let \(a^{1/n} = t\). Raise both sides to power \(n\): \[ (a^{1/n})^n = t^n \Rightarrow a = t^n. \] So \(t\) is an \(n\)-th root of \(a\), giving \(a^{1/n} = \sqrt[n]{a}\).

Checkpoints

Pause and reason before continuing. Discuss with a classmate or write your reasoning in a notebook.

Applications

Large and small scales in science

Distances in astronomy and sizes of atoms are usually written with powers of 10 to keep numbers readable. For example, \(1.5 \times 10^{11}\) m is much clearer than writing 150000000000 m.

Repeated growth and decay

Compound growth often uses exponent models such as \(A = P(1+r)^t\). Exponents track repeated multiplication over equal time steps.

Computing and binary powers

Computer memory sizes are tied to powers of 2. For example, \(2^{10} = 1024\), so one kibibyte is 1024 bytes and larger units build from further powers.

References

• Descartes, R. (1637). La Geometrie (superscript power notation usage).
• Wallis, J. (1685). A Treatise of Algebra (negative and fractional powers).
• Secondary school algebra curricula on laws of indices and scientific notation.