Laws of Exponents

For any numbers b and c and integers m and n:

• b^mbⁿ = b^m + n
• ${\displaystyle \frac{b^m}{b^n}=b^{m - n}}$
• (b^m)ⁿ = b^mn
• b^m c^m = (bc)^m

TIP

Memorize the laws of exponents. They come up quite often on the Math 1 test.

For example,

$\begin{array}{cccccc}{4}^5\times 4^3=4^8\hfill & & & & & (4^5{)}^3=4^{15}\hfill \\ {\displaystyle \frac{4^5}{4^3}}=4^2\hfill & & & & & (4^2\times 4^3{)}^4=(4^5{)}^4=4^{20}\hfill \\ {\displaystyle \frac{4^3}{4^5}}=4^{-2}={\displaystyle \frac{1}{4^2}}& & & & & 3^5\times 4^5={12}^5\hfill \end{array}$

TIP

Be sure to use parentheses. If you need the value of –2 raised to the 6th power and enter –2⁶ on your calculator, you will get –64. Use parentheses: (–2)⁶ = 64.

EXAMPLE 2: ­If 4^m × 4ⁿ = 4¹⁰, what is the average (arithmetic mean) of m and n?

Since 4^m × 4ⁿ = 4^{m + n}:

$4^{10}=4^{m + n}\Rightarrow m + n=10\Rightarrow \frac{m + n}{2}=5$