Laws of exponents

When two powers share the same base, you should never multiply them out the long way. There are short rules that do the job in a single step. They were discovered by counting factors carefully and they hold for any base , counting numbers, fractions, even negative numbers.

Concept

Law 1 , Product of same bases. $\quad a^m \cdot a^n = a^{m+n}$.

Why? $a^m$ has $m$ copies of $a$, $a^n$ has $n$ copies; multiplying them places all of them in one row, $m+n$ in total. Example: $2^3 \cdot 2^4 = (2 \cdot 2 \cdot 2)(2 \cdot 2 \cdot 2 \cdot 2) = 2^7 = 128$.

Law 2 , Quotient of same bases. $\quad \dfrac{a^m}{a^n} = a^{m-n}$ (for $a \ne 0$).

In the quotient $\frac{a^m}{a^n}$, $n$ of the factors in the top cancel with the $n$ factors in the bottom. What remains is $m - n$ factors on top. Example: $\frac{5^7}{5^4} = 5^3 = 125$.

Law 3 , Power of a power. $\quad (a^m)^n = a^{mn}$.

$(a^m)^n$ means "multiply $a^m$ by itself $n$ times", which gives $n \times m$ copies of $a$. Example: $(3^2)^4 = 3^8$.

Law 4 , Power of a product. $\quad (ab)^n = a^n b^n$.

Multiplying $(ab)$ by itself $n$ times rearranges into $n$ copies of $a$ and $n$ copies of $b$. Example: $(2 \cdot 5)^3 = 10^3 = 1000$ and also $2^3 \cdot 5^3 = 8 \cdot 125 = 1000$. ✓

Law 5 , Power of a quotient. $\quad \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$ (for $b \ne 0$).

Same idea as Law 4, but with division.

These five laws compose. If you see $\left(\dfrac{2^3 \cdot 5}{4}\right)^2$, you just use the laws step by step.

A common trap: the laws only combine same bases. The expression $2^3 \cdot 3^4$ does not simplify to a single power , you have to either compute it as $8 \cdot 81 = 648$, or leave it as $2^3 \cdot 3^4$. There is no rule $a^m \cdot b^m = (ab)^m$ unless the exponents match (Law 4), and no rule for unequal bases.

Another trap: $(a+b)^n \ne a^n + b^n$ in general. For example $(1+2)^2 = 9$ but $1^2 + 2^2 = 5$.

Worked examples

Example 1. Simplify $3^4 \cdot 3^5$.

• Same base, add exponents: $3^{4+5} = 3^9$.

Example 2. Simplify $\dfrac{2^8 \cdot 2^3}{2^5}$.

• Top: $2^{8+3} = 2^{11}$.
• Divide: $2^{11-5} = 2^6 = 64$.

Example 3. Simplify $\big((-2)^3\big)^2 \cdot (-2)^4$.

• $\big((-2)^3\big)^2 = (-2)^{6}$.
• Multiply: $(-2)^{6} \cdot (-2)^4 = (-2)^{10} = 1024$.

Example 4. Simplify $\left(\dfrac{2^3 \cdot 3^2}{6}\right)^2$.

• $6 = 2 \cdot 3$, so $\dfrac{2^3 \cdot 3^2}{6} = \dfrac{2^3 \cdot 3^2}{2 \cdot 3} = 2^{2} \cdot 3^{1} = 12$.
• Square: $12^2 = 144$.

Try it yourself

1. Simplify $5^6 \cdot 5^3$ and write as a single power.
2. Simplify $\dfrac{7^9}{7^5}$.
3. Simplify $\big(2^3\big)^4$ and find its value.
4. Simplify $\big(3 \cdot 5\big)^4$ using a single base.
5. Show $(a^2 b^3)^4 = a^8 b^{12}$.
6. Without computing, decide which is greater: $2^{10}$ or $10^2$.
7. Simplify $\dfrac{2^{15} \cdot 3^{10}}{6^{10}}$.
8. Is $3^4 \cdot 5^4 = (15)^4$? Justify.

Activity / Insight

Powers of $2$ chart. Write the powers of $2$ from $2^0 = 1$ up to $2^{20}$. Notice that $2^{10} = 1024 \approx 10^3$. This single approximation, $2^{10} \approx 10^3$, is why "kilobyte" used to mean $1024$ bytes and why doubling something ten times multiplies it by roughly a thousand. From there, $2^{20} \approx 10^6$ (a "mega") and $2^{30} \approx 10^9$ (a "giga"). The laws of exponents and the laws of computing meet here.