Exponents

Laws of Exponents

Exponents have laws and ignorantia juris non excusat.

Law	Property	Example
Zero Power	$x^0=1$	$3^0=1$
One Power	$x^1=x$	$3^1=3$
Negative Exponent	$x^{-m}={\displaystyle \frac{1}{x^m}}$	$3^{-4}={\displaystyle \frac{1}{3^4}}$
Product	$x^m\cdot x^n=x^{m+n}$	$3^4\cdot 3^2=3^6$
Quotient	${\displaystyle \frac{x^m}{x^n}}=x^{m-n}$	${\displaystyle \frac{3^4}{3^2}}=3^2$
Power of Power	$(x^m{)}^n=x^{m\cdot n}$	$(3^4{)}^2=3^8$
Power of Product	$(x\cdot y{)}^m=x^m\cdot y^m$	$(3\cdot 5{)}^4=3^4\cdot 5^4$
Power of Quotient	${\left({\displaystyle \frac{x}{y}}\right)}^m={\displaystyle \frac{x^m}{y^m}}$	${\left({\displaystyle \frac{3}{5}}\right)}^4={\displaystyle \frac{3^4}{5^4}}$

Other examples

Exponents that are 0 equal 1:

$x^0=1$

Exponents that are 1 equal the value of the base:

$x^1=x$

Reciprocate negative exponents to make them positive:

$x^{-7}={\displaystyle \frac{1}{x^7}}$

Add exponents with the same base when multiplied:

$x^3\cdot x^2=x^{3+2}=x^5$

Subtract exponents with the same base when divided:

${\displaystyle \frac{x^5}{x^2}}=x^{5-2}=x^3$

Multiply exponents when raised by the power of another exponent:

$(x^2{)}^3=x^{2\cdot 3}=x^6$

Factor exponents to make them easier to solve:

$3^4=(3^2{)}^2=9^2=81$

Distribute the power when multiplied by the power including the coefficient:

$(2x^2{)}^3=(2^1{x}^2{)}^3=(2^3\cdot x^{2\cdot 3})=(8\cdot x^6)=8x^6$

Watch the placement of parentheses when dealing with negative bases:

$-3^2=-(3\cdot 3)=-9$

$(-3{)}^2=(-3\cdot -3)=9$

Citations, References, and Links