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.
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 \(a^0 = 1\).
Again from the quotient law: \[ \frac{a^0}{a^n} = a^{0-n} = a^{-n}. \] Since \(a^0 = 1\), we get \(a^{-n} = \dfrac{1}{a^n}\).
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 \(a^{1/n} = \sqrt[n]{a}\).
Pause and reason before continuing. Discuss with a classmate or write your reasoning in a notebook.
Checkpoint
Simplify \(x^7 \div x^3\). Which index law did you use?
Checkpoint
Rewrite \(5^{-2}\) without a negative index, then evaluate.
Checkpoint
Simplify \((a^2b^{-1})^3\) and express your final answer with positive indices.
Checkpoint
Evaluate \(81^{3/4}\) by rewriting it in root form first.
Checkpoint
Write \(0.000072\) in scientific notation and state the index on 10.
Checkpoint
True or false: \((x+y)^2 = x^2 + y^2\). Explain why index laws do or do not support this.
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.
Compound growth often uses exponent models such as \(A = P(1+r)^t\). Exponents track repeated multiplication over equal time steps.
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.