Properties of Exponents

when multiplying two powers that have the same base, you can add the exponents

 $a^m\times a^n=a^{m+n}$   

Example: $\left(x^3\right)\left(x^5\right)\ =\ x^{3+5}=x^8$  

Why?  $\left(x.x.x\right)\left(x.x.x.x.x\right)$  

You can divide two powers with the same base by subtracting the exponents.

 $\frac{a^m}{a^n}=a^{m-n}$  

Example:  $\frac{x^{11}}{x^4}=x^{11-4}=x^7$  

Why? $\frac{x.x.x.x.x.x.x.x.x.x.x}{x.x.x.x}=x.x.x.x.x.x.x$  

When a power is raised to a power, just multiply the exponents

 $\left(a^m\right)^n=a^{m.n}$  

Example:  $\left(x^2\right)^6=x^{12}$  

Why?  $\left(x^2\right)^6=\left(x.x\right)\left(x.x\right)\left(x.x\right)\left(x.x\right)\left(x.x\right)\left(x.x\right)$  

When a few factors share the same exponent, distribute the exponent to each factor in the product

 $\left(ab\right)^m=a^mb^m$  

Example:  $\left(x^2y\right)^3$  

Why?  $\left(x^2y\right)^3=\left(x.x.y\right)\left(x.x.y\right)\left(x.x.y\right)=\left(x.x.x.x.x.x.y.y.y\right)=x^6y^3$  

When raising a quotient (fraction to a power), distribute the exponent to both the numerator and denominator in the quotient.

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

Example:  $\left(\frac{x^2}{y}\right)^3$  

Why?  $\left(\frac{x^2}{y}\right)^3=\left(\frac{x.x}{y}\right)\left(\frac{x.x}{y}\right)\left(\frac{x.x}{y}\right)=\left(\frac{x.x.x.x.x.x}{y.y.y}\right)=\frac{x^6}{y^3}$  

Any non-zero base raised to the zero exponent is ALWAYS equals to 1

 $a^0=1$  

Example:  $5^0=1$  

Example:  $\left(-2\right)^0=1$  

Example:  $8x^0=8\times x^0=8\times1=8$  

Any nonzero number raised to a negative power equals its reciprocal raised to the opposite positive power.

as the saying goes "cross the line, change the sign (of the exponent)"

 $a^{-m}=\frac{1}{a^m}$  or  $a^m=\frac{1}{a^{-m}}$  

Example:  $2^{-2}=\frac{1}{2^2}=\frac{1}{4}$