An exponent of a number shows how many times the number is multiplied by itself. In other words, it tells us the power of that number. For example, 44 means 4 × 4 × 4 × 4 = 256. Exponent rules help simplify calculations with numbers raised to a power. Some key rules include multiplying powers with the same base, dividing powers with the same base, raising a power to another power, and applying powers to a product or a quotient. Other important rules are the zero exponent, negative exponent, and fractional exponent rules. The a^m + a^n formula is often used when working with powers of the same base in expressions. Understanding these rules makes it easier to solve problems, simplify complex expressions, and perform calculations quickly and accurately in mathematics.

What are Exponent Rules?

Exponents are notations that denote the times a number is to be multiplied by itself. The different exponent rules help in simplifying the numbers with powers involving decimals, fractions, large power, roots, etc. The exponents can be a fraction, whole numbers, decimals or even negative numbers. In an expression say \(9^{3}\)= 9 × 9 × 9, 3 is the exponent that shows the number of times the number 9 is multiplied. However the number 9 is the base here, that is the number which is getting multiplied.

Exponents Definition

An exponent is the small number written above and to the right of a base number, showing how many times the base is multiplied by itself. In simple words, it tells us how many times to use the number in multiplication. For example, 4³ means 4 × 4 × 4 = 64. This is read as "four to the power of three." When the exponent is 2, it is called "squared," and when it is 3, it is called "cubed." Squared is used in calculating area, while cubed is used for volume.

Exponents are very useful for writing very large or very small numbers. For example, the distance from the Earth to the Sun can be written as 1.496 × 10⁸ km, and the mass of a proton as 1.67 × 10⁻²⁷ kg.

Laws of Exponents Formulas

Exponents follow simple rules that make it easier to work with numbers raised to a power. These rules include adding exponents when multiplying like bases, subtracting exponents when dividing like bases, multiplying exponents when raising a power to another power, applying exponents to each term in a product or quotient, using zero and negative exponents, and handling fractional exponents as roots of numbers.

Product Rule: When multiplying numbers with the same base, add the exponents.
Formula: a^m × a^n = a^(m+n)

Quotient Rule: When dividing numbers with the same base, subtract the exponents.
Formula: a^m / a^n = a^(m-n)

Power of a Power Rule: When raising a power to another power, multiply the exponents.
Formula: (a^m)^n = a^(m×n)

Power of a Product Rule: When multiplying different bases with the same exponent, apply the exponent to each base.
Formula: (a×b)^n = a^n × b^n

Power of a Quotient Rule: When dividing different bases with the same exponent, apply the exponent to both numerator and denominator.
Formula: (a/b)^n = a^n / b^n

Zero Exponent Rule: Any non-zero number raised to 0 equals 1.
Formula: a^0 = 1

Negative Exponent Rule: A negative exponent means taking the reciprocal of the number.
Formula: a^(-n) = 1 / a^n

Fractional Exponent Rule: A fraction as an exponent means taking the root.
Formula: a^(1/n) = n√a and a^(m/n) = n√(a^m)

Laws of Exponents

Exponent rules are also understood as the properties of exponents or laws of exponents that further make the procedure of simplifying expressions involving complex exponents easier. The different laws are as follows:

• Powers with Same Base
• Quotient with Same Base
• Power of a Power
• Product to a Power
• Quotient to a Power
• Zero Power Rule
• Negative Exponent Rule
• Fractional Exponent Rule

Let us learn about each of them in detail, in the below section.

Also learn about Like Fractions and Unlike Fractions here.

Powers Exponent Rules

This rule is also called the product rule of exponents. It says that in the product of two different powers (m,n) of the same number (a), we get (m+n)th power of that number i.e the exponents are added when the bases of the numbers are the same. The rule is commonly used for the multiplication of terms in an expression. The formula for the same is; \(a^m\times a^n=a^{m+n}\) Here, m and n are both real numbers.

Example: 2^3 × 2^4 = 2^(3+4) = 2^7 = 128

Quotient Exponent Rule

The law is also called the quotient law of exponents. When we have the ‘m’th power of a number divided by ‘n’th power of the same number we get ‘(m-n)’th power of that number i.e., exponents are subtracted if the bases of the terms are the same. The rule is based on the dividing expressions with the same bases.

Both the product and quotient law helps in solving the multiplication and division operations in exponents without actually solving the expressions, given that the bases are the same. \(\frac{a^m}{a^n}=a^{m-n}\) Here, both m and n are integers and a is a non-zero term.

Learn how to perform the Long Division process.

Example: 5^6 / 5^2 = 5^(6−2) = 5^4 = 625

Power of a Power Exponent Rule

When a single number(the base) is raised to two exponents, then we use the power of a power rule. The power rule for exponents says that for an individual base multiply the exponents. Mathematically; \(\left(a^m\right)^n=a^{mn}\) Here, both m and n are integers and a is a non-zero term.

Example: (3^2)^4 = 3^(2·4) = 3^8 = 6561

Product Exponent Rule

The rule says that when two or more different numbers(bases) are multiplied with one another such that their powers are the same, then we get the product of the two numbers raised to the common power. \(a^n\times b^n=\left(ab\right)^n\). Here, both m and n are integers and a is a non-zero term.

Example: 2^3 × 5^3 = (2·5)^3 = 10^3 = 1000

Quotient to a Power Exponent Rule

The rule says that in a fraction when the numerator and the denominator are different but their powers are the same then we raise the entire fraction to the common power. \(\frac{a^n}{b^n}=\left(\frac{a}{b}\right)^n\). Here, both m and n are integers and a is a non-zero term.

Example: 4^2 / 2^2 = (4 / 2)^2 = 2^2 = 4

Read more about the Fraction to Percent conversion here.

Zero Power Exponent Rule

The rule says that, when the power of any given integer is zero, then its value is equivalent to 1. It should be noted that the base should not be zero. If the base is zero then 0^{0} is not defined. Mathematically saying; \(a^{0}=1\) Here a is a non-zero term.

Example: 7^0 = 1

Negative Exponent Rule

The rule is used when the exponent or power is negative. It says that if the exponent or the raised power is negative, we can modify the exponent into positive by taking the reciprocal of the given value. That is to convert a negative exponent into a positive exponent the term is transferred to the denominator from the numerator and with this changes the sign of the exponent values.

Mathematically; \(a^{-n}=\frac{1}{a^n}\)

Example: 2^−3 = 1 / 2^3 = 1 / 8

Fractional Exponent Rule

When a fraction is the exponent of a number, it is said to be a fractional exponent. The rule says that; \(a^{\frac{1}{m}}=\sqrt[m]{a}\) For the above expression, a stands for the base, and 1/m is the exponent in the fractional form. \(a^{\frac{m}{n}}=\sqrt[n]{\left(a^m\right)}\).

Example 1: 8^(1/3) = ∛8 = 2

Example 2: 27^(2/3) = ∛(27^2) = ∛729 = 9

Check out this article on Power Set.

Combining Exponents

When dealing with numbers or terms that have different bases, you cannot add or subtract the exponents. The rules for exponents only work when the base of the numbers is the same. If the bases are different, each term must be simplified individually before doing any further calculation.

For example:

• 2² × 3³ cannot be written as 6⁵ because the bases 2 and 3 are different.
• Instead, calculate each term separately: 2² = 4, 3³ = 27. Then multiply the results: 4 × 27 = 108.

Other Rules of Exponents

Besides the main exponent rules, some key points to remember are: negative numbers raised to even or odd powers, any power of 1 is 1, and numbers greater than 1 raised to infinity become infinitely large.

1. Negative Numbers: If a negative number is raised to an even power, the result is positive. If it is raised to an odd power, the result is negative.
   Example: (-3)² = 9, (-3)³ = -27
2. One as a Base: Any power of 1 is always 1.
   Example: 1⁵ = 1, 1¹⁰⁰ = 1
3. Numbers Raised to Infinity: Any number greater than 1 raised to an infinitely large power becomes infinity.
   Example: 3^∞ = ∞

Exponent Rules Chart

Exponent rules help simplify calculations with powers. They include the zero exponent rule (any number to the power 0 is 1), identity rule (any number to the power 1 is itself), product and quotient rules (adding or subtracting powers when multiplying or dividing), negative exponent rule (reciprocals), and power rules (raising a power to another power, product, or quotient). This chart makes learning these rules easy.

Common Mistakes to Avoid in Exponent Rules

Many students make errors with exponents, such as adding exponents with different bases, misunderstanding zero exponents, misusing negative exponents, or incorrectly distributing exponents over products and quotients. Careful application avoids these mistakes.

Adding Exponents with Different Bases – A common mistake is adding exponents when multiplying numbers with different bases. For example, a^m × b^n ≠ a^(m+n). Exponents can only be added if the bases are the same.

Zero Exponent Confusion – Remember that any non-zero number raised to the power of 0 is always 1, never 0. For example, 5^0 = 1.

Incorrect Use of Negative Exponents – A negative exponent does not make a number negative. It means taking the reciprocal of the number. For example, a^(-m) = 1 / a^m.

Improper Distribution of Exponents – When applying an exponent to a product or quotient, make sure to distribute it correctly. For example, (ab)^m = a^m b^m and (a/b)^m = a^m / b^m.

Solved Examples on Exponent Rules

Well acknowledged with the various laws of exponents and related mathematical formulas let us go through some solved examples for more practice.

Solved Example 1: Simplify the expressions below.

\(3^{13}\times3^{15}=?\)

\(\left(-4\right)^6\times\left(-4\right)^8=?\)

Solution: Starting with the first expression that is;

\(3^{13}\times3^{15}=?\)

As the bases are the same using; \(a^m\times a^n=a^{m+n}\)

\(3^{13}\times3^{15}=3^{28}\)

\(\left(-4\right)^6\times\left(-4\right)^8=?\)

Similarly, for the above expression the base 4 is common in both the terms, thus applying the same rule.

\(\left(-4\right)^6\times\left(-4\right)^8=\left(-4\right)^{14}\)

Solved Example 2: If \(3^n\times5^n=15^5\), then determine the value of n.

Solution: Given, \(3^n\times5^n=15^5\)

As per the rules when two terms carry the same power and they are present in product form then the power can be handled as common. The formula for the same is \(a^n\times b^n=\left(ab\right)^n\).

Thus the equation modifies to;

\(15^n=15^5\)

Now when two bases are the same, then their powers can be equated. Thus n=5 here.

Solved Example 3: State the correct name of the rules below.

\(\frac{a^m}{a^n}=a^{m-n}\)

\(a^n\times b^n=\left(ab\right)^n\)

\(a^{0}=1\)

\(a^{-n}=\frac{1}{a^n}\)

Solution: For all the above 4 expression the names for the rules are as follows;

\(\frac{a^m}{a^n}=a^{m-n}\) denotes the quotient law of exponents.

\(a^n\times b^n=\left(ab\right)^n\) specifies the power of product rule of exponents.

\(a^{0}=1\) is the zero law of exponents.

\(a^{-n}=\frac{1}{a^n}\) specifies the negative law of exponents.

Learn the different Divisibility rules here.

Solved Example 4: Solve the below expression and obtain the answer.

\(\frac{4^6}{4^3}\)

\(6^{-4}\)

\(\left(y^6\right)^2\)

\(\left(ab\right)^4\)

Solution: Let us start answering them one by one;

\(\frac{4^6}{4^3}\)=?

Using the rule; \(\frac{a^m}{a^n}=a^{m-n}\)

\(\frac{4^6}{4^3}=4^{6-3}=4^3\)

\(6^{-4}\)=?

Using the rule; \(a^{-n}=\frac{1}{a^n}\)

\(6^{-4}=\frac{1}{6^4}\)

\(\left(y^6\right)^2\)=?

Using the rule; \(\left(a^m\right)^n=a^{mn}\)

\(\left(y^6\right)^2=y^{12}\)

\(\left(ab\right)^4\)

Using the rule; \(a^n\times b^n=\left(ab\right)^n\)

\(\left(ab\right)^4=a^4\times b^4\)

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