Laws of Exponents – Laws, Proofs & Examples

The exponent of a number is a way to show how many times the number is multiplied by itself. We use positive exponents to write very large numbers and negative exponents to write very small numbers. There are some rules that are used to simplify the problems involving exponents.

Let’s understand what are the laws of exponents with proofs and examples.

What are the Laws of Exponents?

The laws of exponents also known as properties of exponents are used to solve problems involving exponents. These properties are also considered major exponents rules. 

The following are the basic properties of exponents.

• Exponent Law of Product: $a^m \times a^n = a^{m+n}$
• Exponent Law of Quotient: $\frac{a^m}{a^n} = a^{m-n}$
• Law of Zero Exponent: $a^0 = 1$
• Law of Negative Exponent: $a^{-m} = \frac{1}{a^m}$
• Exponent Law of Power of a Power: $(a^m)^n = a^{mn}$
• Exponent Law of Power of a Product: $(ab)^m = a^mb^m$
• Exponent Law of Power of a Quotient: $\left(\frac{a}{b} \right)^m = \frac{a^m}{b^m}$
• Exponent with Fractional Index $x^{1}{n} = \sqrt[n]{x}$

1. Exponent Law of Product 

The exponent law of product states that if $a^m$ and $a^n$ are two exponents then the product of $a^m$ and $a^n$ is equal to $a^{m + n}$, i.e. when exponents of the same bases are multiplied then the resultant exponent has power that is equal to the sum of powers of the exponents in the product.

The exponent law of product can be written as $a^m \times a^n = a^{m + n}$.

Proof of Exponent Law of Product 

$a^m = a \times a \times a \times a … \times a (m \text{ times})$

$a^n = a \times a \times a \times a … \times a (n \text{ times})$

$a^m \times a^n = (a \times a \times a \times a … \times a (m \text{ times})) \times (a \times a \times a \times a … \times a (n \text{ times}))$

$=> a^m \times a^n = a \times a \times a \times a … \times a ((m + n) \text{ times})$

$=> a^m \times a^n = a^{m + n}$

Examples on Exponent Law of Product 

Example 1: Evaluate $3^4 \times 3^3$

$3^4 \times 3^3 = 3^{4 + 3} = 3^7$

Example 2: Evaluate $(xy)^5 \times (xy)^6$

$(xy)^5 \times (xy)^6 = (xy)^{5 + 6} = (xy)^11$

Example 3: Evaluate $2^5 \times 2^2 \times 2^7$

$2^5 \times 2^2 \times 2^7 = 2^{5 + 2 + 7} = 2^14$

2. Exponent Law of Quotient

The exponent law of product states that if $a^m$ and $a^n$ are two exponents then the quotient of $a^m$ and $a^n$ is equal to $a^{m – n}$, i.e. when exponents of the same bases are divided then the resultant exponent has power that is equal to the difference of powers of the exponents in the product.

The exponent law of quotient can be written as $a^m \div a^n = a^{m – n}$.

Proof of Exponent Law of Quotient 

$a^m = a \times a \times a \times a … \times a (m \text{ times})$

$a^n = a \times a \times a \times a … \times a (n \text{ times})$

Consider $m > n$, then $a^m \div a^n = \frac{a^m}{a^n}$

$= \frac{a \times a \times a \times a … \times a (m \text{ times})}{a \times a \times a \times a … \times a (n \text{ times})}$

$= \frac{a \times a \times a \times a … \times a ((m – n) \text{ times}) \times a \times a \times a \times a … \times a (n \text{ times}) }{a \times a \times a \times a … \times a (n \text{ times})}$

$= a \times a \times a \times a … \times a ((m – n) \text{ times})$

$=> a^m \div a^n = a^{m – n}$

Examples on Exponent Law of Quotient

Example 1: Evaluate $7^5 \div 7^2$

$7^5 \div 7^2 = \frac{7^5}{7^2} = 7^{5 – 2} = 7^3$

Example 2: Evaluate $(mn)^8 \div (mn)^3$

$(mn)^8 \div (mn)^3 = \frac{(mn)^8}{(mn)^3} = (mn)^{8 – 3} = (mn)^5$

3. Law of Zero Exponent

The exponent law of product states that the value of an exponent is $1$ if its power is $0$. 

The law of zero exponent can be written as $a^0 = 1$.

Proof of Law of Zero Exponent 

$a^m \div a^m = \frac{a^m}{a^m} = 1$ —————– (1) (A number divided by itself results in $1$)

Also $\frac{a^m}{a^m} = a^{m – m}$ ——————— (2) (Exponent Law of Quotient)

From (1) and (2)

$a^{m – m} = 1 => a^0 = 1$

Examples on Law of Zero Exponent

Example 1: Evaluate $6^4 \times 6^2 \times 6^{-6}$

$6^4 \times 6^2 \times 6^{-6} = 6^{4 + 2 + (-6)}$

$6^{4 + 2 + (-6)} = 6^{4 + 2 – 6} = 6^0 = 1$

Example 2: Evaluate $2^0 + 3^0 + 4^0 + 5^0$

$2^0 + 3^0 + 4^0 + 5^0 = 1 + 1 + 1 + 1 = 4$

4. Law of Negative Exponent

The law of negative exponent states that $a^{-m}$ equals the reciprocal of $a^m$.

The law of negative exponent can be written as $a^{-m} = \frac{1}{a^m}$.

Proof of Law of Negative Exponent

$a^{-m} = a^{0 – m} = \frac{a^0}{a^m}$ (Exponent Law of Quotient)

$\frac{a^0}{a^m} = \frac{1}{a^m}$

Therefore $a^{-m} = \frac{1}{a^m}$

Examples on Law of Negative Exponent

Example 1: Evaluate $7^4 \div 7^9$

$7^4 \div 7^9 = \frac{7^4}{7^9} = 7^{4 – 9}$

$= 7^{-5} = \frac{1}{7^5}$

Example 2: Evaluate $x^2y^7 \div x^6y^4$

$x^2y^7 \div x^6y^4 = \frac{x^2y^7}{x^6y^4}$

$= \frac{x^2}{x^6} \times \frac{y^7}{y^4}$

$= x^{2 – 6} \times y^{7 – 4}$

$= x^{-4} \times y^{3}$

$= \frac{1}{x^4} \times y^3 = \frac{y^3}{x^4}$

5. Exponent Law of Power of a Power

The exponent law of the power of power states that when an exponent is again raised to another power, the power of the resulting exponent is the product of two powers.

The exponent law of the power of power can be written as $(a^m)^n = a^{mn}$.

Proof of Exponent Law of Power of a Power

$(a^m)^n = a^m \times a^m \times a^m …. \times a^m (n \text{ times})$

$= (a \times a \times a … \times a (m \text{ times})) \times (a \times a \times a … \times a (m \text{ times})) \times $

$(a \times a \times a … \times a (m \text{ times})) … \times (a \times a \times a … \times a (m \text{ times})) (n \text{ times})$

$= a \times a \times a … \times a (mn \text{ times}) = a^{mn}$

Examples on Exponent Law of Power of a Power

Example 1: Evaluate $(2^3)^2$

$(2^3)^2 = 2^{3 \times 2}$

$= 2^6 = 64$

Example 2: Evaluate $(5^8)^0$

$(5^8)^0 = 5^{8 \times 0}$

$5^0 = 1$

6. Exponent Law of Power of a Product

The exponent law of the power of a product states that the value of an exponent of a product is equal to the product of individual exponents.

The exponent law of the power of a product can be written as $(ab)^m = a^m b^m$.

Proof of Exponent Law of Power of a Product

$(ab)^m = (ab) \times (ab) \times (ab) … \times (ab) (m \text{ times})$

$= (a \times b) \times (a \times b) \times (a \times b) … \times (a \times b) (m \text{ times})$

$= (a \times a \times a … \times a (m \text{ times})) \times (b \times b \times b … \times b (n \text{ times}))$

$= a^m \times b^m $

Examples on Exponent Law of Power of a Product

Example 1: Evaluate $12^4$

$12^4 = (4 \times 3)^4$

$=4^4 \times 3^4$

$= (2^2)^4 \times 3^4$

$= 2^8 \times 3^4$

$= 256 \times 81 = 20736$

7. Exponent Law of Power of a Quotient

The exponent law of the power of a quotient states that the value of an exponent of a quotient is equal to the quotient of individual exponents.

The exponent law of the power of a quotient can be written as $\left(\frac{a}{b} \right)^m = \frac{a^m}{b^m}$.

Proof of Exponent Law of Power of a Quotient

$\left(\frac{a}{b} \right)^m = \frac{a}{b} \times \frac{a}{b} \times \frac{a}{b} … \times \frac{a}{b} (m \text{ times})$ 

$= \frac{a \times a \times a … \times a (m \text{ times})}{b \times b \times b … \times b (m \text{ times})}$

$= \frac{a^m}{b^m}$

Examples on Exponent Law of Power of a Quotient

Example 1: $\left( \frac{5}{2} \right)^3$

$\left( \frac{5}{2} \right)^3 = \frac{5^3}{2^3}$

$= \frac{125}{8}$

Example 2: $\left( \frac{x^2y}{xy^2} \right)^3$

$\left( \frac{x^2y}{xy^2} \right)^3 = \frac{(x^2y)^3}{(xy^2)^3}$

$= \frac{x^{2 \times 3} y^3}{x^3 \times y^{2 \times 3}}$

$= \frac{x^6 y^3}{x^3 \times y^6}$

$= \frac{x^6}{x^3} \times \frac{y^3}{y^6}$

$= x^{6 – 3} \times y^{3 – 6}$

$= x^3 \times y^{-3}$

$= x^3 \times \frac{1}{y^3} = \frac{x^3}{y^3}$

8. Exponent with Fractional Index

Fractional exponents are also called rational exponents. In fractional exponent, the numerator is the power to which the number should be taken and the denominator is the root which should be taken.

The exponent law exponent with fractional index can be written as $a^{1}{n} = \sqrt[n]{a}$

Examples on Exponent with Fractional Index

Example 1: Evaluate $(\sqrt{5})^8$

$(\sqrt{5})^8 = (5^{\frac{1}{2}})^8$

$= 5^{\frac{1}{2} \times 8}$

$= 5^4 = 625$

Example 2: Evaluate $\frac{4}{2 \sqrt{2}}$

$\frac{4}{2 \sqrt{2}} = \frac{2^2}{2 \times 2^{\frac{1}{2}}}$

$=\frac{2^2}{2^1 \times 2^{\frac{1}{2}}}$

$=\frac{2^2}{2^{1 + \frac{1}{2}}}$

$= \frac{2^2}{2^{\frac{3}{2}}}$

$= 2^{2 – \frac{3}{2}}$

$= 2^{\frac{1}{2}} = \sqrt{2}$

Practice Problems

1. Evaluate the following
   • $\left(\frac{1}{2} \right)^{-3}$
   • $\left(\frac{2}{5} \right)^{-2}$
   • $(-3)^{-5}$
   • $\left(\frac{-3}{7}\right)^{-4} \times \left(\frac{-2}{3}\right)^2$
   • $\left(\frac{-1}{2}\right)^{-3} \times \left(\frac{-1}{2}\right)^{-2}$ 
   • $\left(\frac{1}{3} \times \frac{1}{4} \right)^{-1} \div 2^{-1}$
   • $\left(\frac{1}{4} \right)^{-2} + \left(\frac{1}{2} \right)^{-2} + \left(\frac{1}{5} \right)^{-2}$
2. By what number should $\left(\frac{1}{3} \right)^{-1}$ be multiplied so that the product is $\left(-\frac{3}{4} \right)^{-1}$?

FAQs

Conclusion

The laws of exponents are the rules that define the process of simplification of exponents. There are eight basic rules of exponents. These are

• Exponent Law of Product: $a^m \times a^n = a^{m+n}$
• Exponent Law of Quotient: $\frac{a^m}{a^n} = a^{m-n}$
• Law of Zero Exponent: $a^0 = 1$
• Law of Negative Exponent: $a^{-m} = \frac{1}{a^m}$
• Exponent Law of Power of a Power: $(a^m)^n = a^{mn}$
• Exponent Law of Power of a Product: $(ab)^m = a^mb^m$
• Exponent Law of Power of a Quotient: $\left(\frac{a}{b} \right)^m = \frac{a^m}{b^m}$
• Exponent with Fractional Index $x^{1}{n} = \sqrt[n]{x}$

Recommended Reading

• Difference Between Exponent and Power