Laws of Exponents

Laws of Exponents

Try reading this - 27,00,00,00,00,00,00,00,00,000. How hard is it? A massive number that cannot be read right? Reading natural numbers that are massive is a hard task.  Not just that, they are difficult to recognize as well as evaluate. Now, to easily read these numbers, we tend to convert them into exponents. Exponents are nothing but powers or indices of a number. In this article, you will be learning about what are exponents? What is the key to the laws of exponents? What are the rules of exponents? Some solved examples to understand this concept better and frequently asked questions to summarize everything you learn.

Laws of Exponents Definition

The power of a number is known as the exponent. It is usually expressed as a raised number or raised symbol. For example, 37, where 7 is the exponent, and it can also be written as \[3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3.\]

Laws of Exponent Formula

1. rm rn = r ( m + n )

Example: 

35 32 = 3( 5 + 2 )

35 32 = 37

35 32 = \[3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3.\]

35 32 = 2187

2. \[\;{\left( {{\text{ }}{r^m}} \right)^n} = {\text{ }}{r^{mn}}\]

Example: 

( 34 )2 = \[{3^4} \times 2\]

( 34 )2 = 38

( 34 )2 =  \[3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3\]

( 34 )2 = 6561

3. \[\;{\left( {rs} \right)^m} = {r^m}{s^m}\]

Example: 

( 2 \[ \times \] 4 )5 = 25 \[ \times \] 45

( 2 \[ \times \] 4 )5 = \[2 \times 2 \times 2 \times 2 \times 2 \times 4 \times 4 \times 4 \times 4 \times 4\]

( 2 \[ \times \] 4 )5 = 32,768

4. \[{\left( {\frac{r}{s}} \right)^n} = \left( {\frac{{{r^n}}}{{{s^n}}}} \right)\]

Example: (\[\frac{3}{5}\])⁶ = (\[\frac{3^{6}}{5^{6}}\] 

(\[\frac{3}{5}\])⁶ =(\[\frac{3\times3\times3\times3\times3\times3}{5\times5\times5\times5\times5\times5}\])

(\[\frac{3}{5}\])⁶ = (\[\frac{243}{3125}\])

5. \[\frac{{{r^n}}}{{{r^m}}} = {r^{n - m}}\]

Example: \[\frac{4^{12}}{4^{4}}\] = \[4^{12・4}\]

\[\frac{4^{12}}{4^{4}}\] = \[4^{8}\]

\[\frac{4^{12}}{4^{4}}\] = 4 X 4 X 4 X 4 X 4 X 4 X 4 X 4

\[\frac{4^{12}}{4^{4}}\] = 65536

6. \[\frac{{{r^n}}}{{{r^m}}} = \frac{1}{{{r^{m - n}}}}\]

Example: \[\frac{7^{6}}{7^{10}}\] = \[\frac{1}{7^{10-6}}\]

\[\frac{7^{6}}{7^{10}}\] = \[\frac{1}{7^{4}}\]

\[\frac{7^{6}}{7^{10}}\] =\[\frac{1}{7 \times 7 \times 7 \times 7}\]

\[\frac{7^{6}}{7^{10}}\] = \[\frac{1}{2401}\]

Rules for Laws of Exponents with Examples

Exponents are nothing but the power of a number. They tell how many times the number is repeated. In other words, these are the numbers that are raised to the power of another number. With the rules stated below, there are laws of exponents examples too. The following specialized names are used to represent the numbers:

1. 23 = “ 2 is raised to the power 3” or “ 2 cubed”

2. 42 = “ 4 is raised to the power 2” or “ 4 squared”

Example: 10,00,000 = \[10 \times 10 \times 10 \times 10 \times 10 \times 10 = {10^6}\]

To simplify the expressions, you need to follow some rules. The rules to solve these expressions are:

Rule 1: When the bases are the same and the operation between the numbers is multiplication, then the powers of these numbers are just added. 

rm \[ \times \] rn = r ( m + n )

Here, r is the base, and m and n are the exponents. 

Example: 

25 22 = 2( 5 + 2 )

25 22 = 27

25 22 = \[2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\]

25 22 = 128

Rule 2: When a number is raised by another number, the powers are multiplied with each other.

\[{\left( {{\text{ }}{r^m}} \right)^n} = {\text{ }}{r^{m{\text{ }} \times {\text{ }}n}}\]

Here, r is the base and m and n are the exponents. 

Example: 

( 54 )2 = \[{5^{4 \times 2}}\]

( 54 )2 = 58

( 54 )2 =  \[5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5\]

( 54 )2 = 390625

Rule 3: When two different numbers have the same base, each base will be raised to the same power. 

( rs )m = rm sm

Here, r and s are the base, and m is the exponent. 

Example: 

( 1 \[ \times \] 2 )5 = 15 \[ \times \] 25

( 1 \[ \times \] 2 )5 = \[1 \times 1 \times 1 \times 1 \times 1 \times 2 \times 2 \times 2 \times 2 \times 2\]

( 1 \[ \times \] 2 )5 = 32

Rule 4: When the bases are different and the operation between the bases is division, then the powers of these bases are raised to each base individually. 

\[{\left( {\frac{r}{s}} \right)^n} = \left( {\frac{{{r^n}}}{{{s^n}}}} \right)\]

Here, r and s are the base, and m is the exponent. 

Example:  (\[\frac{3}{4}\])⁶ = (\[\frac{3^{6}}{4^{6}}\])

(\[\frac{3}{4}\])⁶ =(\[\frac{3\times3\times3\times3\times3\times3}{4\times4\times4\times4\times4\times4}\])

(\[\frac{3}{4}\])⁶ = (\[\frac{243}{4096}\])

Rule 5: When the bases are the same and the operation between the bases is division, then the power of the denominator is subtracted from the power of the numerator. 

\[\frac{{{r^n}}}{{{r^m}}} = {r^{n - m}}\]

Here, r is the base, and m and n are the exponents. 

Example:  \[\frac{2^{12}}{2^{4}}\] = \[2^{12・4}\]

\[\frac{2^{12}}{2^{4}}\] = \[2^{8}\]

\[\frac{2^{12}}{2^{4}}\] = 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2

\[\frac{2^{12}}{2^{4}}\] = 256

Rule 6: When the bases are the same and the operation between the bases is division, then the power of the numerator is subtracted from the power of the denominator. 

\[\frac{{{r^n}}}{{{r^m}}} = \frac{1}{{{r^{m - n}}}}\]

Here, r is the base, and m and n are the exponents. 

Example:  \[\frac{3^{4}}{3^{10}}\] = \[\frac{1}{3^{10-4}}\]

\[\frac{3^{4}}{3^{10}}\] = \[\frac{1}{7^{6}}\]

\[\frac{3^{4}}{3^{10}}\] =\[\frac{1}{3 \times 3 \times 3 \times 3}\]

\[\frac{3^{4}}{3^{10}}\] = \[\frac{1}{729}\]