Table of Contents
Last modified on November 22nd, 2024
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.
Here, the exponent is a positive integer.
It is mathematically written as:
Here are a few more examples of positive integer exponents:
Here, the exponent is a negative integer. It is the reciprocal of the base raised to the positive exponent.
It is mathematically written as:
Here are a few more examples of negative integer exponents:
In this case, the exponent is 0, and the base is a non-zero number.
It is, mathematically, written as:
Here, are some more examples: 130 = 1, and 80 = 1
Here is a summary of the different types of integer exponents.
Here is a list of the laws followed while solving expressions involving integer exponents:
Product Property | xm ⋅ 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 property | x0 = 1 |
One Exponent Property | x1 = x |
Derivative Property | ${\dfrac{d}{dx}\left( x^{n}\right) =n\cdot x^{n-1}}$ |
Integral Property | ${\int x^{n}dx=\dfrac{x^{n+1}}{n+1}+C}$ |
Identify the base and the exponent in the given expressions:
a) 5-4
b) -113
c) (-3)7
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
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) ${\dfrac{1}{8^{-2}}}$
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
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 ${\dfrac{\left( 3^{2}x^{3}\right) ^{2}\cdot \left( \dfrac{y^{-1}}{9x^{-2}}\right) ^{3}}{\left( 3xy^{2}\right) ^{-2}}}$
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 November 22nd, 2024