Table of Contents
In this chapter we assume you already know what an exponent is and how to read expressions like $2^3$ or $10^4$. Here we focus on the rules (laws) that tell you how exponents behave when you combine them.
Throughout, think of a simple base like $a$ and an exponent like $m$ or $n$ where $a$ is any nonzero number and $m,n$ are whole numbers (we’ll hint at zero and negative exponents but leave full treatment to other chapters).
The product rule: multiplying same bases
When you multiply powers with the same base, you add the exponents:
$$
a^m \cdot a^n = a^{m+n}.
$$
Example:
- $2^3 \cdot 2^4 = 2^{3+4} = 2^7$.
- $10^2 \cdot 10^3 = 10^{2+3} = 10^5$.
Why this works (for positive whole exponents):
$2^3$ means $2 \cdot 2 \cdot 2$ and $2^4$ means $2 \cdot 2 \cdot 2 \cdot 2$.
Together you have $7$ factors of $2$, so $2^7$.
Key point: same base, multiply → add exponents.
The quotient rule: dividing same bases
When you divide powers with the same base, you subtract the exponents:
$$
\frac{a^m}{a^n} = a^{m-n}
$$
as long as $a \neq 0$.
Example:
- $\dfrac{3^5}{3^2} = 3^{5-2} = 3^3$.
- $\dfrac{10^6}{10^4} = 10^{6-4} = 10^2$.
Reason (for $m \ge n$):
$\dfrac{3^5}{3^2} = \dfrac{3\cdot3\cdot3\cdot3\cdot3}{3\cdot3}$.
Cancel two $3$’s from top and bottom, leaving $3^3$.
Key point: same base, divide → subtract exponents (top minus bottom).
Power of a power: raising a power to a power
When you raise a power to another power, you multiply the exponents:
$$
(a^m)^n = a^{m \cdot n}.
$$
Example:
- $(2^3)^4 = 2^{3\cdot 4} = 2^{12}$.
- $(10^2)^3 = 10^{2\cdot 3} = 10^6$.
Reason:
$(2^3)^4$ means $(2^3)\cdot(2^3)\cdot(2^3)\cdot(2^3)$, which is four factors of $2^3$.
Using the product rule, $2^3\cdot 2^3\cdot 2^3\cdot 2^3 = 2^{3+3+3+3} = 2^{12}$.
Key point: power of a power → multiply exponents.
Power of a product: multiplying bases then raising to a power
When a product is raised to a power, the exponent applies to each factor:
$$
(ab)^n = a^n b^n.
$$
Example:
- $(2 \cdot 5)^3 = 10^3 = 1000$, and also $2^3 \cdot 5^3 = 8 \cdot 125 = 1000$.
- $(3x)^2 = 3^2 x^2 = 9x^2$ (here $x$ is a variable).
Key point: exponent “distributes” over multiplication.
Be careful: this only works cleanly with multiplication, not with addition. For example, generally $(a+b)^2 \neq a^2 + b^2$.
Power of a quotient: dividing bases then raising to a power
When a fraction is raised to a power, the exponent applies to numerator and denominator:
$$
\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
$$
for $b \neq 0$.
Example:
- $\left(\dfrac{2}{3}\right)^2 = \dfrac{2^2}{3^2} = \dfrac{4}{9}$.
- $\left(\dfrac{5x}{2}\right)^3 = \dfrac{5^3 x^3}{2^3} = \dfrac{125x^3}{8}$.
Key point: exponent “distributes” over division as well.
Zero exponent (idea only)
Using the quotient rule, consider
$$
\frac{a^m}{a^m} = a^{m-m} = a^0.
$$
But also $\dfrac{a^m}{a^m} = 1$ (any nonzero number divided by itself).
So for $a \neq 0$ it is consistent to define
$$
a^0 = 1.
$$
Example:
- $2^0 = 1$, $10^0 = 1$, $(-3)^0 = 1$.
Full treatment and edge cases (like $0^0$) belong in another chapter; here you just need the basic rule: any nonzero number to the zero power equals 1.
Negative exponents (idea only)
Again from the quotient rule, for $m < n$:
$$
\frac{a^m}{a^n} = a^{m-n}.
$$
For instance,
$$
\frac{a^2}{a^5} = a^{2-5} = a^{-3}.
$$
But also
$$
\frac{a^2}{a^5} = \frac{1}{a^3}.
$$
So it is consistent to define
$$
a^{-n} = \frac{1}{a^n}, \quad a \neq 0.
$$
Example:
- $2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$.
- $10^{-2} = \dfrac{1}{10^2} = \dfrac{1}{100}$.
Full work with negative exponents will be developed elsewhere; here the key idea is that a negative exponent represents a reciprocal.
Summary of the main laws
For a nonzero number $a$ and whole numbers $m,n$:
- Product rule (same base):
$$
a^m \cdot a^n = a^{m+n}.
$$ - Quotient rule (same base, $a \neq 0$):
$$
\frac{a^m}{a^n} = a^{m-n}.
$$ - Power of a power:
$$
(a^m)^n = a^{mn}.
$$ - Power of a product:
$$
(ab)^n = a^n b^n.
$$ - Power of a quotient ($b \neq 0$):
$$
\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}.
$$ - Zero exponent (for $a \neq 0$):
$$
a^0 = 1.
$$ - Negative exponent (for $a \neq 0$; preview):
$$
a^{-n} = \frac{1}{a^n}.
$$
These laws let you simplify and manipulate expressions with exponents systematically, and they will be used repeatedly in later topics.