Exponent Rules: Definition, Laws, Examples

Many arithmetic tasks such as addition, subtraction, multiplication, and division can be easily done in quick steps using exponent rules. These rules also help to simplify numbers with complex forces including fractions, decimals, and roots. The various rules and regulations of exponents are also known as exponents laws.

Key terms: Power, Exponent, Product law, quotients, Law of exponents, zero exponents, Negative law, Power Exponents, Quotient Law of Exponents

Also read: Isosceles Triangle Theorems

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What are the Exponent Laws?

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Laws of exponents, also known as exponent rules or properties of exponents, helps in making complex multiplication expressions simple and reduce the number of steps during solving a sum. These rules help to simplify decimal expressions, fractions, irrational numbers, and negative integers as their exponents.

For example, if we need to solve 3⁴ × 3², we can easily do it using one of the object rules, a^m × aⁿ = a^{m + n}. Using this rule, we will simply add exponents to get the answer, while the basics are still the same, i.e., 3⁴ × 3² = 3^{4 + 2} = 3⁶. Similarly, high-value exponential statements can be easily resolved with help of exponent rules.

Also read: Circles 

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Rules for Exponents

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The various rules and regulations of exponents are also known as exponents laws. A few of them are as follows: 

• Product law of exponents
• Quotient law of exponents
• Law of zero exponents
• Negative law of exponents
• The Law of Power Exponents
• Power of Product Exponents Law
• The Power of the Quotient Law of Exponents.

Also read: Tangent to a Circle

Product Law for Exponents

• The Product law is used to replicate expressions that have the same base value. 
• The rule states, "To repeat the process of multiplication with the same base, add the exponents while the base remains constant." In simpler terms, this rule involves adding powers with the same base. 
• Here the law is useful to simplify two terms with the function of repetition between them.
• a^m × aⁿ = a^m+n
• Example: 2³ × 2⁶ = 2³⁺⁶ =2⁹

Also read: Areas Related to Circles

Quotient Law of Exponents

• The quotient property of the objects is used to separate the expressions involving the same base.
• This property states, "To divide two expressions with the same base value, subtract their exponents while the base remains constant." 
• The only condition required is that the dividend and divisor must have the same bases.
• a^m/aⁿ = a^m-n
• Example: 5⁴/5² = 5^4-2 = 5²

Also read: Eccentricity

Zero Exponent Law

• This rule is applicable only when the exponent raised to the base value is 0. 
• This rule states, "Any number (other than 0) raised from 0 is 1." 
• Note that 00 is not defined. This will help us to understand that regardless of the base the value of any number raised to 0 will be equal to 1.
• a⁰ = 1
• Example: 7⁰ = 1

Also read: Eccentric Formula

Identity Exponent Law

• Any number raised to an exponent of one is equal to the number itself.
• a¹ = a
• Example: 4¹ = 4

Negative Law of Exponents

• This rule is applicable only when the given exponent is a negative number. 
• The property states, "To convert any negative exponent into a positive exponent, it must be divided upon one (reciprocal)." 
• This way the numerator becomes the denominator with a change in the sign.
• a^-m = 1/a^m or (a/b)^-m = (b/a)^m
• Example: 7^-6 = 1/7⁶

Also read: Eccentricity

The Law of Power Exponents

• This rule is applicable to expressions of form (am) n. 
• The rule states, "If we have one base with two exponents, just multiply the powers." 
• Two exponents are found with one on top of the other. These can be easily replicated to form a single element.
• (a^m)ⁿ = a^mn
• Example: (5⁴)² = 5^4*2 = 5⁸

Power of Product Exponents Law

• The power of product exponents rule is analogous to the distributive property in mathematics. 
• Consider an expression with two base values together raised to an exponent, then the expression can be simplified by writing the base values separately with the same exponent value.
• (ab)^m = a^m x b^m
• Example: (2*5)⁴ = 2⁴ x 5⁴

Also read: Integers As Exponents?

The Power of the Quotient Law of Exponents

• The property of the power of quotient law works the same way as that of power of product exponents, except that the base value is expressed in the form of a division.
• (a/b)^m = a^m/b^m
• Example: (4/5)² = 4²/5²

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Tips on Laws of Exponents

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• If a fraction has a negative exponent, then we take the reciprocal of the fraction to make the exponent positive: (a/b)^-m = (b/a)^m
• We can convert an expression with roots into an exponent using the following property: a¹/n = n√a

Also read: Ordinate

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Things to remember

• Laws of exponents, also known as exponent rules or properties of exponents, help in making complex multiplication expressions simple and reduce the number of steps during solving a sum.
• A few of the rules are as follows: Product law of exponents, Law of quotients, Law of exponents, Law of zero exponents, Negative law of exponents, The Law of Power Exponents, Power of Product Exponents Law, and The Power of the Quotient Law of Exponents.