Exponent Vocabulary and Laws

Exponent Vocabulary and Laws

Coefficient

A number multiplied by a variable or variables.
The coefficient is 4:      $4x$  
The coefficient is -6:      $-6a^2b$  
The coefficient is 1/2:      $\frac{1}{2}pq^3t^5$  

Exponent

The number of times a base (number or variable) is used as a factor.
EX: $\ a\cdot a\cdot a\cdot a\cdot a\ \ =\ a^5$           "a" is used as a factor 5 times. 

So 5 is the exponent. 

Base

The number being raised to a power.
EX:  In the expression, 

 $12^4$  the base is 12.

Multiple Choice

In the expression 15⁸, what is the base?

• 8

• 15

Multiple Choice

In the expression 15⁸, what is the exponent?

• 8

• 15

Forms:

Exponential Form:   $6^3$  

Expanded Form:  $6\cdot6\cdot6$  
Standard Form (answer):  $216$  

Multiple Choice

Which of the following is in exponential form?

• 625

• $5\cdot5\cdot5\cdot5$

• 25(15+10)

• $5^4$

Multiple Choice

Which of the following is in expanded form?

• 625

• $5\cdot5\cdot5\cdot5$

• 25(10+15)

• $5^4$

Multiple Choice

Which of the following is in standard form?

• 625

• $5\cdot5\cdot5\cdot5$

• 25(10+15)

• $5^4$

Exponent Rules

Product Rule: Multiply Coefficients, keep the base, add the exponents. 

 $2x^3\cdot6x^5=\left(2\cdot6\right)x^{\left(3+5\right)}=12x^8$  
Quotient Rule: Divide Coefficients, keep the base, subtract exponents.
 $\frac{9a^5}{3a^2}=\left(\frac{9}{3}\right)a^{\left(5-2\right)}=3a^3$  

Multiple Choice

Simplify: $3x^5\cdot4x^2$  

•  $12x^7$  

•  $7x^{10}$  

•  $12x^{10}$  

•  $7x^7$  

Multiple Choice

Simplify:  $\frac{15x^4y^4}{3x^3y}$  

•  $12xy^4$  

•  $18x^7y^5$  

•  $5xy^3$  

•  $5x^7y^4$  

Exponent Rules

Power Rule: The base stays the same, multiply the exponents
EX: $\left(x^3\right)^4=x^{\left(3\cdot4\right)}=x^{12}$  
Power of a Product: Raise each factor in parenthesis to the power.
EX: $\left(3a^3bc^5\right)^2=\left(3\right)^2\cdot\left(a^3\right)^2\cdot\left(b\right)^2\cdot\left(c^5\right)^2=9a^6b^2c^{10}$ 

 Power of a Quotient: Raise each factor in parenthesis to the power.

EX:  $\left(\frac{2x^2}{y^4z^8}\right)^6=\frac{\left(2\right)^6\cdot\left(x^2\right)^6}{\left(y^4\right)^6\left(z^8\right)^6}=\frac{64x^{12}}{y^{24}z^{48}}$  

Multiple Choice

 $What\ numbers\ should\ go\ in\ place\ of\ the\ \Delta\ and\ O?$   $\left(4x^4\right)^2=\Delta\ x^O$  

• 16, 16

• 16, 8

• 8, 8

• 8, 6

Multiple Choice

 $Simplify:\ \ \left(\frac{4a^7}{b^5c^4}\right)^3=$  

•  $\frac{12a^{10}}{b^8c^7}$  

•  $\frac{16a^{12}}{b^{20}c^{20}}$  

•  $\frac{64a^{21}}{b^{15}c^{12}}$  

Exponent Rules

 Zero Power: anything, except 0, raised to the zero power equals 1.
EX: $14^0=1$                 $B^0=1$                   $\left(x^4y^9\right)^0=1$  
Negative Power: take the reciprocal of the base and make the exponent positive.

EX:   $5^{-2}=\frac{1}{5^2}=\frac{1}{25}$           $3x^{-4}y^2=\frac{3y^2}{x^4}\left(only\ the\ x\ had\ the\ negative\ \exp\right)$  

Multiple Choice

 $\left(7n^3m^2p^6\right)^{0\ }=$  

• 0

• 1

•  $7n^3m^2p^6$  

Multiple Choice

Simplify: $\frac{6^0x^3y^{-2}}{z^{-4}}=$  

•  $\frac{6x^3}{y^2z^4}$  

•  $\frac{x^3z^4}{y^2}$  

•  $xyz^4$