Exponent Rules

Exponent rules are those laws that are used for simplifying expressions with exponents. Many arithmetic operations like addition, subtraction, multiplication, and division can be conveniently performed in quick steps using the laws of exponents. These rules also help in simplifying numbers with complex powers involving fractions, decimals, and roots.

Let us learn more about the different rules of exponents, involving different kinds of numbers for the base and exponents.

Exponent rules, which are also known as the 'laws of exponents' or the 'properties of exponents' make the process of simplifying expressions involving exponents easier. These rules are helpful to simplify the expressions that have decimals, 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 exponent rules which says, a^m × aⁿ = a^{m + n}. Using this rule, we will just add the exponents to get the answer, while the base remains the same, that is, 3⁴ × 3² = 3^{4 + 2} = 3⁶. Similarly, expressions with higher values of exponents can be conveniently solved with the help of the exponent rules. Here is the list of exponent rules.

• a⁰ = 1
• a¹ = a
• a^m × aⁿ = a^m+n
• a^m / aⁿ = a^m−n
• a^−m = 1/a^m
• (a^m)ⁿ = a^mn
• (ab)^m = a^mb^m
• (a/b)^m = a^m/b^m

The different rules of exponents are also known as the laws of exponents (or) properties of exponents. The laws of exponents were already mentioned in the previous section. Most of them have specific names such as the product rule of exponents, the quotient rule of exponents, the zero rule of exponents, the negative rule of exponents, etc.

Let us learn each of these in detail now.

The product rule of exponents is used to multiply expressions with the same bases. This rule says, "To multiply two expressions with the same base, add the exponents while keeping the base the same." This rule involves adding exponents with the same base. Here the rule is useful to simplify two expressions with a multiplication operation between them.

Observe the following example.

This shows that without using the law, the expression involves more calculation.

The quotient law of exponents is used to divide expressions with the same bases. This rule says, "To divide two expressions with the same base, subtract the exponents while keeping the base same." This is helpful in solving an expression, without actually performing the division process. The only condition that is required is that the two expressions should have the same base.

Here is an example.

We can clearly see that without using the law, the expression involves more calculation.

The zero law of exponents is applied when the exponent of an expression is 0. This rule says, "Any number (other than 0) raised to 0 is 1." Note that 0⁰ is not defined, it is an indeterminate form. This will help us understand that irrespective of the base the value for a zero exponent is always equal to 1.

Here is an example.

Using the law we simply get 2⁰ = 1. Alternatively, without using the law we can understand the same law with more number of steps.

The negative law of exponents is used when an exponent is a negative number. This rule says, "To convert any negative exponent into positive exponent, the reciprocal should be taken." The expression is transferred from the numerator to the denominator with the change in sign of the exponent values.

Here is an example.

Using the law, we can solve it in just one go, like 2^-2 = 1/(2²). Alternatively, without using the law the process is lengthy.

The 'power of a power law of exponents' is used to simplify expressions of the form (a^m)ⁿ. This rule says, "When we have a single base with two exponents, just multiply the exponents." The two exponents are available one over the other. These can be conveniently multiplied to make a single exponent.

Here is an example.

Using this law reduces the process of calculation.

The 'power of a product rule of exponents' is used to find the result of a product that is raised to an exponent. This law says, "Distribute the exponent to each multiplicand of the product."

Here is an example.

Using the law, (xy)³ = x³.y³. On the other hand, the same thing can be expressed in multiple steps, without using the law. (xy)³ =(xy).(xy).(xy) = (x.x.x).(y.y.y) x³.y³

The power of a quotient rule of exponents is used to find the result of a quotient that is raised to an exponent. This law says, "Distribute the exponent to both the numerator and the denominator." Here, the bases are different and the exponents are the same for both the bases.

Here is an example of the exponent rule given above.

We can use the law and simply solve, and we can also solve the same expression without the law which involves multiple steps.

The fractional exponents rule says, a^1/n = ⁿ√a. i.e., When we have a fractional exponent, it results in radicals. For example, a^1/2 = √a, a^1/3 = ∛a, etc. This rule is further extended for complex fractional exponents like a^m/n. Using the power of a power rule of exponents (that we have studied in one of the previous sections),

a^m/n = (a^m)^1/n

Now, by using the fractional exponents rule, this fractional power turns into a radical.

a^m/n = ⁿ√(a^m)

This is also used as an alternate form of the fractional exponents rule. Thus, this rule is defined in two ways:

• a^1/n = ⁿ√a
• a^m/n = ⁿ√(a^m)

The rules of exponents explained above can be summarized in a chart as shown below.

Tips on Exponent Rules:

• If a fraction has a negative exponent, then we take the reciprocal of the fraction to make the exponent positive, i.e., (a/b)^-m = (b/a)^m
• We can convert a radical into an exponent using the following rule: a^1/n = ⁿ√a

☛ Related Articles

• Exponential Equations
• Irrational Exponents
• Exponent Rules Calculator

What are Exponent Rules in Math?

Exponent rules are those laws which are used for simplifying expressions with exponents. These laws are also helpful to simplify the expressions that have decimals, fractions, irrational numbers, and negative integers as their exponents. For example, if we need to solve 34⁵ × 34⁷, we can use the exponent rule which says, a^m × aⁿ = a^m+n, that is, 34⁵ × 34⁷ = 34^{5 + 7} = 34¹² . A few rules of exponents are listed as follows:

• Product Rule: a^m × aⁿ = a^m+n;
• Quotient Rule: a^m/aⁿ = a^m-n;
• Negative Exponents Rule: a^-m = 1/a^m;
• Power of a Power Rule: (a^m)ⁿ = a^mn.

What are the 8 Laws of Exponents?

The 8 laws of exponents can be listed as follows:

• Zero Exponent Law: a⁰ = 1
• Identity Exponent Law: a¹ = a
• Product Law: a^m × aⁿ = a^m+n
• Quotient Law: a^m/aⁿ = a^m-n
• Negative Exponents Law: a^-m = 1/a^m
• Power of a Power: (a^m)ⁿ = a^mn
• Power of a Product: (ab)^m = a^mb^m
• Power of a Quotient: (a/b)^m = a^m/b^m

What is the Purpose of the Exponent Rules?

The purpose of exponent rules is to simplify the exponential expressions in fewer steps. For example, without using the exponent rules, the expression 2³ × 2⁵ is written as (2 × 2 × 2) × (2 × 2 × 2 × 2 × 2) = 2⁸. Now, with the help of exponent rules, this can be simplified in just two steps as 2³ × 2⁵ = 2^{(3 + 5)} = 2⁸.

How to Prove the Laws of Exponents?

The exponent laws can be proved easily by expanding the terms. The exponential expression is expanded by writing the base as many times as the power value. The exponent of the form aⁿ is written as a × a × a × a × a × .... n times. Further, on multiplying we can obtain the final value of the exponent. For example, let us solve 4² × 4⁴. Using the 'product law' of exponents, which says a^m × aⁿ = a^m+n, we get 4² × 4⁴ = 4^{2 + 4} = 4⁶. This can be expanded and checked as (4 × 4) × (4 × 4 × 4 × 4) = 4096. We know that the value of 4⁶ is also 4096. Hence, the exponent rules can be proved by expanding the given terms.

What are the Exponent Rules when Bases are the same?

When the bases are the same, all the laws of exponents can be applied. For example, to solve 3¹² ÷ 3⁴, we can apply the 'Quotient Rule' of exponents in which the exponents are subtracted. So, 3¹² ÷ 3⁴ will become 3^12-4 = 3⁸. Similarly, to solve 4⁹ × 4⁴, we apply the 'Product Rule' of exponents in which the exponents are added. This will result in 4⁹⁺⁴ = 4¹³.

What are the Exponent Rules when Bases are Different?

When the bases and powers are different, then each term is solved separately and then we move to the further calculation. For example, let us add 4² + 2⁵ = (4 × 4) + (2 × 2 × 2 × 2 × 2) = 16 + 32 = 48. This process is applicable to addition, subtraction, multiplication, and division. In another example, if the expressions with different bases and different powers are multiplied, each term is evaluated separately and then multiplied. For example, 10³ × 6² = 1000 × 36 = 36000.

What is the Rule for Zero Exponents?

The rule of zero exponents is a⁰ = 1. Here, 'a', which is the base can be any number other than 0. This law says, "Any number (except 0) raised to 0 is 1." For example, 5⁰ = 1, x⁰ = 1 and 23⁰ = 1. However, note that 0⁰ is not defined.

What is the Difference Between Exponents and Powers?

Exponents and powers sometimes are referred to as the same thing. But in general, in the power a^m, 'm' is referred to as an exponent. You can understand the differences in depth by clicking here.

Can the Exponent be a Fraction?

Yes, the exponent value can be a fraction. The exponent rule relating to the fraction exponent value is (a^m)^1/n = a^m/n. This rule is sometimes helpful to simplify and transform a surd into an exponent.