Last modified on October 28th, 2024

chapter outline

 

Integer Exponents

An integer exponent is a mathematical notation used to represent the repeated multiplication of a base number. When a number x is raised to an integer exponent n, it means multiplying x by itself n times. 

It is of the general form xn

Integer exponents can be positive numbers, negative numbers, or even zero.

Positive 

Here, the exponent is a positive integer. 

It is mathematically written as:

For example, 32 means 3 × 3 = 9. Here, the exponent 2 is a positive integer. 

Here are a few more examples of positive integer exponents:

  • 64 = 6 × 6 × 6 × 6 = 1296
  • (-9)4 = (-9) × (-9) × (-9) × (-9) = 6561
  • 113 = 11 × 11 × 11 = 1331

Negative 

Here, the exponent is a negative integer. It is the reciprocal of the base raised to the positive exponent. 

It is mathematically written as:

For example, 3-2 = ${\dfrac{1}{3}\times \dfrac{1}{3}}$ or ${\dfrac{1}{9}}$

Here are a few more examples of negative integer exponents:

  • 6-4 = ${\dfrac{1}{6}\times \dfrac{1}{6}\times \dfrac{1}{6}\times \dfrac{1}{6}}$ = ${\dfrac{1}{1296}}$
  • 7-3 =  ${\dfrac{1}{7}\times \dfrac{1}{7}\times \dfrac{1}{7}}$ = ${\dfrac{1}{343}}$
  • (-4)-2 = ${\dfrac{1}{-4}\times \dfrac{1}{-4}}$ or ${\dfrac{1}{16}}$

Zero 

In this case, the exponent is 0, and the base is a non-zero number.

It is, mathematically, written as:

x0 = 1

For example, 20 = 1, 130 = 1, and 80 = 1

Here is a summary of the different types of integer exponents.

Properties

Here is a list of the laws followed while solving expressions involving integer exponents:

Product Propertyxm ⋅ xn = xm + n 
Quotient Property${\dfrac{x^{m}}{x^{n}}=x^{m-n}}$
Power of Product Property(xy)m = xmym
Power of Quotient Property${\dfrac{x^{m}}{y^{m}}=\left( \dfrac{x}{y}\right) ^{m}}$
Power of Power Property(xm)n = xmn
Quotient of Negative Exponent Property${\left( \dfrac{a}{b}\right) ^{-n}=\left( \dfrac{b}{a}\right) ^{n}}$
Zero Exponent propertyx0 = 1
One Exponent Propertyx1 = x
Derivative Property${\dfrac{d}{dx}\left( x^{n}\right) =n\cdot x^{n-1}}$
Integral Property${\int x^{n}dx=\dfrac{x^{n+1}}{n+1}+C}$

Solved Examples

Identify the base and the exponent in the given expressions: 
a) 5-4
b) -113
c) (-3)7

Solution:

a) Given, 5-4 
Here, the base is 5, and the exponent or power is -4
b) Given, -113
Here, the base is 11 (as there are no parentheses that indicate including the minus sign), and the exponent is 3 
c) Given, (-3)7 
Here, the base is -3 (as there is a parenthesis indicating 3 with the minus sign), and the exponent is 7

Evaluate the following exponents:
a) 44
b) 7-3

Solution:

a) Here, 44 
= 4 × 4 × 4 × 4
= 256
b) Here, 7-3 
= ${\dfrac{1}{7}\times \dfrac{1}{7}\times \dfrac{1}{7}}$
= ${\dfrac{1}{343}}$

Simplify the following expressions:
a) (34)2 × (32)
b) 50
c) ${\dfrac{1}{8^{-2}}}$

Solution:

a) Here, (34)2 × (32)
= (38) × (32) [Using the power of power property]
= 38 + 2 [Using product property]
= 310 
b) Here, 50 
= 1 [Using zero exponent property]
c) Here, ${\dfrac{1}{8^{-2}}}$
= ${\left( \dfrac{1}{8}\right) ^{-2}}$ [Using power of quotient property]
= ${\left( \dfrac{8}{1}\right) ^{2}}$ [Using quotient of negative exponent property]
= ${64}$

Expand the following numbers in the exponent form:
a) 8967
b) 633.89

Solution:

a) Given, 8967
= 8 × 1000 + 9 × 100 + 6 × 10 + 7 × 1 
= 8 × 103 + 9 × 102 + 6 × 101 + 7 × 100 
b) Given, 633.89
= 6 × 100 + 3 × 10 + 3 × 1 + 8 × 0.1 + 9 × 0.01
= 6 × 102 + 3 × 101 + 3 × 100 + 8 × 10-1 + 9 × 10-2

Simplify the expression ${\dfrac{\left( 3^{2}x^{3}\right) ^{2}\cdot \left( \dfrac{y^{-1}}{9x^{-2}}\right) ^{3}}{\left( 3xy^{2}\right) ^{-2}}}$

Solution:

Here, ${\dfrac{\left( 3^{2}x^{3}\right) ^{2}\cdot \left( \dfrac{y^{-1}}{9x^{-2}}\right) ^{3}}{\left( 3xy^{2}\right) ^{-2}}}$
= ${\dfrac{\left( 3^{2}\right) ^{2}\cdot \left( x^{3}\right) ^{2}\cdot \dfrac{\left( y^{-1}\right) ^{3}}{\left( 9x^{-2}\right) ^{3}}}{\left( 3\right) ^{-2}\cdot \left( x\right) ^{-2}\cdot \left( y^{2}\right) ^{-2}}}$ [Using power of power property]
= ${\dfrac{3^{4}\cdot x^{6}\cdot \dfrac{y^{-3}}{\left( 9\right) ^{3}\cdot \left( x^{-2}\right) ^{3}}}{3^{-2}\cdot x^{-2}\cdot y^{-4}}}$
= ${\dfrac{3^{4}\cdot x^{6}\cdot \dfrac{y^{-3}}{\left( 3^{2}\right) ^{3}\cdot \left( x^{-2}\right) ^{3}}}{3^{-2}\cdot x^{-2}\cdot y^{-4}}}$ [Using power of power property]
= ${\dfrac{3^{4}\cdot x^{6}\cdot \dfrac{y^{-3}}{3^{6}\cdot x^{-6}}}{3^{-2}\cdot x^{-2}\cdot y^{-4}}}$
= ${\dfrac{3^{4-6}\cdot x^{6-\left( -6\right) }\cdot y^{-3}}{3^{-2}\cdot x^{-2}\cdot y^{-4}}}$ [Using quotient property]
= ${\dfrac{3^{-2}\cdot x^{12}\cdot y^{-3}}{3^{-2}\cdot x^{-2}\cdot y^{-4}}}$ 
= 3-2 – (-2) ⋅ x12 – (-2) ⋅ y-3 – (-4) [Using quotient property]
= 30 ⋅ x14 ⋅ y1 
= x14y [Using zero exponent and one exponent properties]Thus, ${\dfrac{\left( 3^{2}x^{3}\right) ^{2}\cdot \left( \dfrac{y^{-1}}{9x^{-2}}\right) ^{3}}{\left( 3xy^{2}\right) ^{-2}}}$ = x14y

Last modified on October 28th, 2024