In this section, we will learn how to operate with exponents.  

 

Recall that $x^5=x\cdot x\cdot x\cdot x\cdot x$.  It means $x$ is multiplied 5 times. In this expression, $x$ is the base and $5$ is the exponent.  Based on this definition, we can conduct multiplication and division on exponential expressions. 

 

Multiplications Rules:

  1. $x^n\cdot x^m=x^{m+n}$
  2. $(x^n)^m=x^{n\cdot m}$
  3. $(x\cdot y)^n=x^n\cdot y^n$

 

Example: Perform the given operation using the multiplication properties of exponents and write your answer in simplest form. 

a)     $b^2\cdot b^3=b^{2+3}=b^{5}$

b)     $a^8a^{9}a^{14}=a^{8+9+14}=a^{31}$

c)     $4x^4\cdot 7x^6=(4\cdot 7)(x^4\cdot x^6)=28x^{10}$

d)     $(5x^4y^2)(2x^7y^3=(5\cdot 2)(x^4\cdot x^7)(y^2\cdot y^3)=10x^{11}y^{5}$

e)     $(2x)^3=2^3\cdot x^3=8x^{3}$

f)     $(-4a^2b^5)^3=(-4)^3\cdot (a^2)^3\cdot (b^5)^3=-64\cdot a^{2\cdot 3}\cdot b^{5\cdot 3}=-64a^6b^{15}$

 

Division Rules:

  1. $\dfrac{x^n}{ x^m}=x^{n-m}$
  2. $\left(\dfrac{x}{ y}\right)^n=\dfrac{x^n}{ y^n}$

 

Example: Perform the given operation using the division properties of exponents and write your answer in simplest form. 

a)     $\dfrac{b^8}{ b^7}=b^{8-7}=b^{1}=b$

b)     $\dfrac{x^{12}y^{2}}{x^{8}y}=x^{12-8}y^{2-1}=x^4y^1=x^4y$

c)     $\dfrac{8x^{6}y^{2}}{6x^{5}y^7}=\dfrac{8}{6}\cdot\dfrac{x^6}{x^5}\cdot\dfrac{y^2}{y^7}=\dfrac{4}{3}\cdot\dfrac{x^{6-5}}{1}\cdot\dfrac{1}{y^{7-2}}=\dfrac{4}{3}\cdot\dfrac{x}{y^5}=\dfrac{4x}{3y^5}$

 (Note: $y^5$ is in the denominator because $y$ in the denominator has higher power than  the $y$ in the numerator)

 

The division rules introduce 0 exponent and negative exponents. 

Zero Exponent: 

$a^0=1$

 

Example: Evaluate.

a)     $15^0=1$

b)     $(-15)^0=1$   (Note: the base of the expression is -15)

c)     $-15^0=-1$   (Note: the base of the expression is 15, not -15)

c)      $(15x)^0=1$

 

Negative exponents:

  1. $a^{-n}=\dfrac{1}{a^n}$
  2. $\dfrac{1}{a^{-n}}=a^n$
  3. $\left(\dfrac{a}{ b}\right)^{-n}=\left(\dfrac{b}{ a}\right)^{n}$

Note:  The negative exponent rules can be used to switch terms from numerator to denominator or vice versa, and is useful to write expressions using positive exponents only.

 

Example.  Write the given expression using positive exponents only.

a)     $\dfrac{x^{-3}y^2}{ z^4}=\dfrac{y^2}{x^3 z^4}$

b)     $\dfrac{x^4}{y^{-5} z^4}=\dfrac{x^4y^5}{ z^2}$

 

In this part, we only introduce the basic rules of exponents. More advanced problems will be covered in MAT 1275 (or MAT 1275CO). 

 

Practice Problems: 

1.   Simplify the exponential expressions. 

(a)   $(4x^5y^4)(4x^2y^2)$

(b)   $\dfrac{14a^{20}b^{20}}{7ab^{17}}$ 

(c)   $\left(\dfrac{3}{w^6}\right)^4$

(d)   $(2x^2)^0$    

2.   Write the expression with positive exponents

(a)    $a^5b^{-2}$

(b)    $\dfrac{x^3}{y^{-3}}$

Answer Key: 1(a)         (b)          (c)            (d)              2(a)      (b)         

 

For more detailed explanation, please read: Arithmetic|Algebra Chapter 5.