Laws of Exponents for Real Numbers

Laws of Exponents are fundamental rules used in mathematics to simplify expressions involving exponents. These laws help in solving arithmetic problems efficiently by defining operations like multiplication, division, and more on exponents. In this article, we will discuss the laws of exponent for real number in detail with some solved examples.

What are the Laws of Exponents?

Laws or rules which help us govern the calculation of exponents are called laws of exponent. Laws of Exponents for Real Numbers are:

• Product Law
• Quotient Law
• Power Law
• Power of Product Law
• Power of Quotient Law
• Zero Power Rule
• Exponent of Exponent

Let’s discuss the laws in detail as follows.

Product law

The Product law states that if the base of two numbers is same, their exponents can be added directly.

a^m × aⁿ = a^m+n

To verify this property, see how many times “a” is multiplied in total. It is multiplied “m + n” times. Thus, the property. 

Example: a²a³ = (aa)(aaa) = a⁵

Quotient law

According to this law, if the two numbers in the numerator and denominator are the same, their exponents can be arranged in such a manner that the exponent in the denominator is subtracted by the exponent in the numerator 

[Tex]\frac{a^m}{a^n} = a^{m-n}[/Tex]

This can also be verified similar to the previous one, just see the number of times a is multiplied and then reduce that number by the number of times it is divided. 

Example: [Tex]\frac{x^4}{x^2}  =\frac{xxxx}{xx} = x^{4-2} = x^2[/Tex]

Power Law

According to Power-law, if an exponent is a power of another exponent, we can simply multiply the exponents.

(a^m)ⁿ = a^mn

First, multiply “a” m times, and then do this operation n times. 

Example: (x³)² = (xxx)² = (xxx)(xxx) = x⁶ = x^{(3 × 2)}

Power of Product law

According to the power of product law, if two real numbers say, a and b here are multiplied and raised to power m, we can distribute the exponent to both a and b separately.

a^m x b^m = (ab)^m

This property is just a rearrangement of all these variables. 

Example: a³ × b³ = (aaa) (bbb) = (ab)(ab)(ab) = (ab)³

Power of Quotient law

According to the power of quotient law, if two real numbers are in numerator and denominator and are raised to power n, the power can be separately distributed to both the numbers.

[Tex](\frac{x}{y})^n = \frac{x^n}{y^n}[/Tex]

This also can be verified by a simple rearrangement of the variables.

Example: [Tex]\frac{x^3}{y^3} = \frac{xxx}{yyy} = \frac{x}{y}\frac{x}{y}\frac{x}{y} = (\frac{x}{y})^{3}[/Tex]

Zero Power Rule

As long as x is not equal to zero, raising it to the power of zero should give us 1 as result.

a⁰ = 1, a≠ 1. 

Let’s take an example to make this more evident, 

1 = [Tex]1 = \frac{x^a}{x^a}  = x^{a – a} = x^0[/Tex]

Note:

Expression 0⁰ is considered to be indeterminate. Why? Because there are two answers to it. 

We know, x⁰ = 1, So, 0⁰ = 1. 

We also know, 0^x = 0. So, 0⁰ = 0. 

These are two contradictory answers, thus we consider 0⁰ to be indeterminate. 

Exponent of Exponent

Sometimes in more complex scenarios, exponents over exponents are given. Let’s see how to approach them, 

[Tex]5^{2^{4}}[/Tex]

We will start solving from up, 

[Tex]5^{2^{4}} \\ = 5^{16}[/Tex]

Let’s look at some examples and questions where these properties are applicable. 

Summary: Laws of Exponents for Real Numbers

We will now see the all laws of exponents in a tabular form.

Law	Formula
Product law	[Tex]a^m \times a^n = a^{m+n}[/Tex]
Quotient law	[Tex] \frac{a^m}{a^n} = a^{m-n}[/Tex]
Power Law	[Tex](a^m)^n = a^{m \times n} [/Tex]
Power of Product law	[Tex](a \times b)^n = a^n \times b^n [/Tex]
Power of Quotient law	[Tex]\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} [/Tex]
Zero Power Rule	[Tex]a^0 = 1 [/Tex],[Tex]a \neq 0 [/Tex]
Negative Exponent Rule	[Tex]a^{-n} = \frac{1}{a^n} [/Tex]

Sample Problems: Laws of Exponents

Problem 1: Find the value of 2^-3

Solution:

[Tex]2^{-3}\\ = \frac{1}{2^3}\\ = \frac{1}{8}[/Tex]

Problem 2: [Tex]\frac{1}{3^{-2}}[/Tex]

Solution:

[Tex]\frac{1}{3^{-2}} \\ = 3^{-2} \\ = \frac{1}{3^2}  \\= \frac{1}{9}[/Tex]

Problem 3: Simplify using above properties: 

(-4)⁵ x (-4)^-10

Solution:

(-4)⁵ x (-4)^-10

= -4^(5-10)                     (a^m × aⁿ = a^m+n)

= (-4)^-5

[Tex]= \frac{1}{(-4)^5}\\ \hspace{0.6cm} (a^{-m} = \frac{1}{a^m}) \\ = \frac{1}{-1024}[/Tex]

Problem 4: Simplify using above properties: 

[Tex]2^5 \div 2^{-6}[/Tex]

Solution:

[Tex]2^5 \div 2^{-6} \hspace{0.6cm} (\frac{a^m}{a^n} = a^{m-n})\\ = \frac{2^5}{2^{-6}} \\ = 2^{5 + (-6)} \\ = 2^{-1} \\ = \frac{1}{2}[/Tex]

Problem 5: Simplify, [Tex](x^3 \div x^{\frac{1}{2}}) \times (x^{\frac{3}{2}} \div x^{0}) \times x^7[/Tex]

Solution:

[Tex](x^3 \div x^{\frac{1}{2}}) \times (x^{\frac{3}{2}} \div x^{0}) \times x^7 \\ = \frac{x^3}{x^{\frac{1}{2}}} \times \frac{x^{\frac{3}{2}}}{x^0} \times x^7 \\ = x^{3 – \frac{1}{2}} \times x^{\frac{3}{2} – 0} \times x^{7} \\ = x^{\frac{5}{2}} \times x^{\frac{3}{2}} \times x^{7} \\ = x^{\frac{5}{2} + \frac{3}{2} + 7}  \\ = x^{\frac{5 + 3+ 14}{2}}  \\ = x^{\frac{22}{2}}\\ = x^{11}[/Tex]

Problem 6: What is the 5th root of 5¹⁵. 

Solution:

We know that nth of a number “a” is represented by [Tex]a^{\frac{1}{n}}[/Tex]

Thus, 5th of 5¹⁵ will be given by, 

[Tex]5^{\frac{15}{5}}[/Tex]

= 5³

= 125

Problem 7: Find the value of n 

[Tex]2^{n-3} = \frac{1}{16^{n-4}} [/Tex]

Solution:

[Tex]2^{n-3} = \frac{1}{16^{n-4}} \\ 2^{n-3} = \frac{1}{2^{4(n-4)}}[/Tex]

⇒ 2^n-3 × 2^{4n – 16} = 1 

⇒ 2^{n – 3 + 4n – 16} = 2⁰

Comparing Both Sides

n – 3 + 4n -16 = 0 

⇒ 5n -19 = 0 

⇒ n =[Tex] \frac{19}{5}[/Tex]