The Timpson Lyceum · Intermediate Algebra (2023)

Lecture 9 (Intermediate Algebra) Properties of Exponents

Overview

This lecture establishes the complete set of properties of exponents and shows where each one comes from rather than asking you to memorize them blindly. Everything follows from two basic ideas: that a positive integer exponent means repeated multiplication, and that a negative exponent means a reciprocal. From those two definitions we derive the product, quotient, power, distribution, zero-exponent, and negative-exponent rules, then practice combining them on increasingly messy expressions. The lecture closes by applying these same rules to scientific notation — writing very large or very small numbers as a single non-decimal digit times a power of ten, and computing with such numbers.

Key Concepts

Two foundational definitions. Every other rule is built from these.

Positive integer exponent (repeated multiplication): $a^n = \underbrace{a \cdot a \cdots a}_{n \text{ factors}}$ . For example, $2^4 = 2\cdot2\cdot2\cdot2 = 16$ .
Negative exponent (reciprocal): $a^{-n} = \dfrac{1}{a^{n}}$ . For example, $2^{-4} = \dfrac{1}{2^{4}} = \dfrac{1}{16}$ .

Product rule (multiplying like bases). Multiplying powers of the same base means concatenating the factors, so you add the exponents.

$a^{m}\cdot a^{n} = a^{m+n}$ . For example, $a^{3}\cdot a^{5} = a^{8}$ .

Quotient rule (dividing like bases). Division is multiplication by a reciprocal, $\dfrac{a^m}{a^n} = a^m \cdot a^{-n}$ , so you subtract the exponents.

$\dfrac{a^{m}}{a^{n}} = a^{m-n}$ . For example, $\dfrac{a^{7}}{a^{4}} = a^{3}$ .

Power of a power. A power raised to a power repeats the inner group, so you multiply the exponents.

$\left(a^{m}\right)^{n} = a^{mn}$ . For example, $\left(a^{2}\right)^{3} = a^{6}$ .

Power of a product (distributing the exponent). An exponent on a product applies to each factor.

$(ab)^{m} = a^{m}b^{m}$ . For example, $(ab)^{3} = a^{3}b^{3}$ .

Power of a quotient. Likewise, an exponent on a fraction applies to numerator and denominator.

$\left(\dfrac{a}{b}\right)^{n} = \dfrac{a^{n}}{b^{n}}$ .

Zero exponent. Anything (nonzero) raised to the zero power equals one, not zero. This follows from $a^{0} = a^{n-n} = \dfrac{a^{n}}{a^{n}} = 1$ .

$a^{0} = 1$ . For example, $10^{0} = 1$ and $(3x^2)^{0} = 1$ . A common error is letting a zero exponent "cancel to zero" — it always gives one.

Moving negative exponents across the fraction bar. A negative exponent in the numerator moves to the denominator (and vice versa) to become positive.

$\dfrac{1}{a^{-n}} = a^{n}$ and $a^{-n} = \dfrac{1}{a^{n}}$ . You may relocate any factor across the bar to flip the sign of its exponent.

Negative exponent on a fraction (flip the fraction). Raising a fraction to a negative power flips it.

$\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^{n} = \dfrac{b^{n}}{a^{n}}$ .

Common trap to avoid. The exponent does not distribute over addition: $(a+b)^{n} \neq a^{n}+b^{n}$ . Check with $(1+2)^2 = 3^2 = 9$ versus $1^2 + 2^2 = 1+4 = 5$ — different values.

Scientific notation. Write a number as a single nonzero digit (the rest decimals) times a power of ten: $d.dddd \times 10^{k}$ .

A large number has a positive exponent (shift the decimal right); a small number has a negative exponent (shift left). For example, $875{,}000 = 8.75\times10^{5}$ and $0.0123 = 1.23\times10^{-2}$ .
To multiply/divide in scientific notation, combine the decimal parts and use the exponent rules on the powers of ten. If a result is no longer in proper form (e.g. $0.53\times10^{4}$ ), borrow a factor of ten: $0.53\times10^{4} = 5.3\times10^{3}$ .

Worked Example

Simplify $\left(\dfrac{3a^{2}\left(2a^{3}b^{2}\right)^{4}}{6a^{5}b^{7}}\right)^{2}$ , applying the rules from the inside out (parentheses, then exponents).

\begin{aligned} \left(\frac{3a^{2}\left(2a^{3}b^{2}\right)^{4}}{6a^{5}b^{7}}\right)^{2} &= \left(\frac{3a^{2}\cdot 2^{4}a^{12}b^{8}}{6a^{5}b^{7}}\right)^{2} &&\text{power of a product: } (a^m)^n=a^{mn} \\ &= \left(\frac{48\,a^{14}b^{8}}{6a^{5}b^{7}}\right)^{2} &&\text{product rule: add exponents } (2+12=14) \\ &= \left(8\,a^{9}b^{1}\right)^{2} &&\text{quotient rule: subtract exponents } (14-5,\;8-7) \\ &= 8^{2}a^{18}b^{2} \;=\; 64\,a^{18}b^{2} &&\text{power of a product again} \end{aligned}

Summary

Two definitions drive everything: $a^{n}$ is repeated multiplication and $a^{-n}=\dfrac{1}{a^{n}}$ .
Multiplying like bases adds exponents; dividing subtracts them; a power of a power multiplies them.
An exponent distributes over products and quotients but never over a sum: $(a+b)^n \neq a^n+b^n$ .
Any nonzero base to the zero power is 1, and negative exponents move across the fraction bar (or flip a fraction) to become positive.
Scientific notation writes a number as one nonzero digit times $10^{k}$ : positive $k$ for large numbers, negative $k$ for small ones.
Multiply and divide in scientific notation by handling the decimal parts and the powers of ten separately, then re-normalize so exactly one digit sits left of the decimal point.