Properties of Exponents

Properties of Exponents (Review)

An overview of the basic exponent properties

Product Rule

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

Multiple Choice

According to the product rule, when when multiplying two powers that have the same base, you can ______the exponents

• add

• subtrct

• multiply

• divide

Multiple Choice

Simplify

 $x^3y^4\left(x^2y^2\right)$  

•  $x^6y^6$  

•  $x^5y^6$  

•  $x^5y^8$  

•  $x^6y^8$  

Quotient Rule

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

Multiple Choice

According to the quotient rule, when we divide two powers with the same base, we can ______ the exponents.

• add

• subtract

• multiply

• divide

Multiple Choice

Simplify

 $\frac{x^7y^5}{xy^2}$  

•  $x^6y^7$  

•  $x^8y^7$  

•  $x^7y^{10}$  

•  $x^6y^3$  

Power Rule

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

Multiple Choice

According to the power rule, when we raise a power to a power, we ___ the exponents.

• add

• subtract

• multiply

• divide

Multiple Choice

Simplify

 $\left(x^4\right)^6$  

•  $x^{10}$  

•  $x^{-2}$  

•  $x^2$  

•  $x^{24}$  

Power of a Product Rule (Distributive Rule)

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

Multiple Choice

Simplify

 $\left(xy^5\right)^2$  

•  $x^2y^{10}$  

•  $x^2y^3$  

•  $xy^3$  

•  $xy^{10}$  

Multiple Choice

Simplify

 $\left(3a^2c\right)^3$  

•  $9a^6c^3$  

•  $27a^5c^3$  

•  $27a^6c^3$  

•  $9a^5c^3$  

Power of a Quotient Rule (Distributive Rule)

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

Multiple Choice

Simplify

 $\left(\frac{x^4}{y^5}\right)^3$  

•  $\frac{x^7}{y^8}$  

•  $\frac{x^{12}}{y^{15}}$  

•  $\frac{x^7}{y^{15}}$  

•  $\frac{x^{12}}{y^8}$  

Zero Exponent Rule

• 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:  $8x^0=8\times x^0=8\times1=8$  

Multiple Choice

What does anything to the zero power equals to?

• 0

• 1

• itself

• the variable

Multiple Choice

Simplify

 $m^0$  

• 1

• 0

• m

• 1m

Negative Exponents

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

Multiple Choice

Simplify

 $x^2y^{-3}$  

•  $\frac{y^{-3}}{x^2}$  

•  $\frac{x^2}{y^3}$  

•  $\frac{y^3}{x^{-2}}$  

•  $\frac{y^3}{x^2}$  

Multiple Choice

Simplify

 $\left(x^3y^{-2}\right)^{-3}$  

•  $\frac{y^6}{x^9}$  

•  $\frac{x^{-9}}{y^5}$  

•  $\frac{x^9}{y^6}$  

•  $\frac{y^6}{x^{-9}}$