Integer Exponents

Exponential Notation

A product of identical expressions can be written using exponential notation.

For example:

$\underset{3 f a c t o r s}{\underset{︸}{7 \cdot 7 \cdot 7}}$ can be written as 7³

$\underset{6 f a c t o r s}{\underset{︸}{(\frac{1}{2}) \cdot (\frac{1}{2}) \cdot (\frac{1}{2}) \cdot (\frac{1}{2}) \cdot (\frac{1}{2}) \cdot (\frac{1}{2})}} can be w r i t t e n a s {(\frac{1}{2})}^{6}$

In general:

If a is any real number and n is a positive integer, then

$\underset{n f a c t o r s}{\underset{︸}{a \cdot a \cdot \dots \cdot a}} = a^{n}$

The number a is called the base, n is called the exponent or power.

aⁿ is called an exponential expression.

Laws of Exponents

Products of Exponential Expressions with the Same Base

Examples

$7^{3} \cdot 7^{2} = \underset{3 f a c t o r s}{\underset{︸}{7 \cdot 7 \cdot 7}} \cdot \underset{2 f a c t o r s}{\underset{︸}{7 \cdot 7}} = \underset{5 f a c t o r s}{\underset{︸}{7 \cdot 7 \cdot 7 \cdot 7 \cdot 7}} = 7^{3 + 2} = 7^{5}$ .

$5^{2} \cdot 5^{4} = \underset{2 f a c t o r s}{\underset{︸}{5 \cdot 5 \cdot}} \cdot \underset{4 f a c t o r s}{\underset{︸}{5 \cdot 5 \cdot 5 \cdot 5}} = \underset{6 f a c t o r s}{\underset{︸}{5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5}} = 5^{2 + 4} = 5^{6}$ .

$a^{n} \cdot a^{m} = \underset{n f a c t o r s}{\underset{︸}{a \cdot \dots \cdot a}} \cdot \underset{m f a c t o r s}{\underset{︸}{a \cdot \dots \cdot a}} = \underset{n + m f a c t o r s}{\underset{︸}{a \cdot \dots a \cdot \dots \cdot a}} = a^{n + m}$ .

Conclusion:

aⁿa^m= a^{n + m}

Quotients of Exponential Expressions with the Same Base

Examples

Conclusion:

$\frac{a^{m}}{a^{n}} = a^{m - n}$

Zero Exponents

Examples

Conclusion:

a⁰=1

Negative Exponents

Examples

Conclusion:

$\frac{1}{a^{n}} = a^{- n}$

Exponents Raised to a Power

Examples

${(7^{3})}^{2} = {(7 \cdot 7 \cdot 7)}^{2} = \underset{2 g r o u p s o f f a c t o r s}{\underset{︸}{\underset{3 f a c t o r s}{\underset{︸}{(7 \cdot 7 \cdot 7)}} \cdot \underset{3 f a c t o r s}{\underset{︸}{(7 \cdot 7 \cdot 7)}}}} = \underset{3 \cdot 2 = 6 f a c t o r s}{\underset{︸}{7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7}} = 7^{3 \cdot 2}$ .

${(a^{m})}^{n} = {(\underset{m f a c t o r s}{\underset{︸}{a \cdot \dots \cdot a}})}^{n} = \underset{n g r o u p s o f f a c t o r s}{\underset{︸}{\underset{m f a c t o r s}{\underset{︸}{(\underset{m f a c t o r s}{\underset{︸}{a \cdot \dots \cdot a}})}} \cdot \dots \cdot \underset{m f a c t o r s}{\underset{︸}{(\underset{m f a c t o r s}{\underset{︸}{a \cdot \dots \cdot a}})}}}} = \underset{m \cdot n f a c t o r s}{\underset{︸}{a \cdot a \cdot \dots \cdot a}} = a^{m n}$ .

Conclusion:

(a^m)ⁿ = a^mn

Products Raised to a Power

Examples

${(3 \cdot 2)}^{4} = \underset{4 f a c t o r s}{\underset{︸}{(3 \cdot 2) \cdot (3 \cdot 2) \cdot (3 \cdot 2) \cdot (3 \cdot 2)}} = \underset{4 f a c t o r s}{\underset{︸}{(3 \cdot 3 \cdot 3 \cdot 3)}} \cdot \underset{4 f a c t o r s}{\underset{︸}{(2 \cdot 2 \cdot 2 \cdot 2)}} = 3^{4} 2^{4}$ .

${(a \cdot b)}^{m} = \underset{m f a c t o r s}{\underset{︸}{(a \cdot b) \cdot \dots \cdot (a \cdot b)}} = \underset{m f a c t o r s}{\underset{︸}{(a \cdot \dots \cdot a)}} \cdot \underset{m f a c t o r s}{\underset{︸}{(b \cdot \dots \cdot b)}} = a^{m} b^{m}$ .

Conclusion:

(ab)^m = a^mb^m

Quotients Raised to a Power

Examples

${(\frac{3}{2})}^{6} = \underset{6 f a c t o r s}{\underset{︸}{(\frac{3}{2}) \cdot (\frac{3}{2}) \cdot (\frac{3}{2}) \cdot (\frac{3}{2}) \cdot (\frac{3}{2}) \cdot (\frac{3}{2})}} = \frac{\overset{6 f a c t o r s}{\overset{︷}{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}}}{\underset{6 f a c t o r s}{\underset{︸}{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}}} = \frac{3^{6}}{2^{6}}$ .

${(\frac{a}{b})}^{m} = \underset{m f a c t o r s}{\underset{︸}{(\frac{a}{b}) \cdot \dots \cdot (\frac{a}{b})}} = \frac{\overset{m f a c t o r s}{\overset{︷}{a \cdot \dots \cdot a}}}{\underset{m f a c t o r s}{\underset{︸}{b \cdot \dots \cdot b}}} = \frac{a^{m}}{b^{m}}$ .

Conclusion:

${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}$