What Is Exponent Arithmetic?

By Robert O

Exponent Arithmetic is like any game that has rules. If you play by the rules, you never get into trouble with the referee. Your work is to master the rules of exponents through continuous practice. Everything will be easy when you know these rules.

Exponent Arithmetic involves working with the bases and the exponents. The representation of exponents is

    \[a^{n}\]

where a is the base and n the exponent. To understand this topic, we will first state all the rules and then wrap it up with solved examples to show how the rules work.

What are the rules and properties of exponents?

Product rules:

If the two number have the same base, then this rule applies:    

    \[a^{n} \times a^{m}=a^{n+m}\]

Example:

    \[3^{2} \times 3^{3}=3^{2+3}=3^{5}=3 \times 3 \times 3 \times 3 \times 3=243\]

If you are working with two numbers of different bases but the same exponents, then this is the rule:

    \[a^{n} \times b^{n}=(a \times b)^{n}\]

Example:

    \[3^{2} \times 2^{2}=(3 \times 2)^{2}=6^{2}=36\]

Quotient rule:

If the bases for the two numbers you are dividing are the same, then the rule is: 

    \[\frac{a^{n}}{a^{m}}=a^{n-m}\]

Example:

    \[\frac{3^{5}}{3^{2}}=3^{5-2}=3^{3}=27\]

If you are dividing two numbers with the same exponent, then the rule is

    \[\frac{a^{n}}{b^{n}}=\left(\frac{a}{b}\right)^{n}\]

.

Example:

    \[\frac{8^{2}}{2^{2}}=\left(\frac{8}{2}\right)^{2}=4^{2}=16\]

Power rule:

We have two power rules as follows.    

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

Example:

    \[\left(2^{2}\right)^{2}=2^{2 \times 2}=2^{4}=16\]

The second power rule is:

    \[a^{n^{m}}=a^{m^{n}}\]

Example:

    \[2^{2^{3}}=2^{3^{2}}=2^{9}=512\]

Power rule with fractional exponents:

    \[a^{\frac{n}{m}}=\sqrt[m]{a^{n}}\]

Example:

    \[27^{\frac{2}{3}}=\sqrt[3]{27^{2}}=9\]

 

 

Negative power rule:

    \[a^{-n}=\frac{1}{a^{n}}\]

           and

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

Example:

    \[3^{-2}=\frac{1}{3^{2}}=\frac{1}{9}\]

Zero rule:

    \[a^{0}=1\]

Example:

    \[7^{0}=1\]

Operation with exponents

Numbers with exponents are also subjects of arithmetic operations. You can add them together, subtract any pair given to you. However, it may not be that direct. Applying the rules of exponential arithmetic, you can easily carry out any operation on such numbers.

How do I add or subtract exponents?

Any operation involving addition or subtraction with exponents require that you first group terms and then solve each group separately.

Example:

    \[\begin{array}{c}a^{2}+3 a^{5}+a^{2}+4 a^{5}+2 a^{3}+7 a^{2}=\left(a^{2}+a^{2}+7 a^{2}\right)+2 a^{3}+\left(3 a^{5}+4 a^{5}\right) \\=9 a^{2}+2 a^{3}+3 a^{5}+4 a^{5}\end{array}\]

The same procedure is applicable when subtracting exponents. Let’s demonstrate using an example:

    \[a^{2}+3 a^{5}-3 a^{2}-4 a^{5}+2 a^{3}+7 a^{2}=\left(a^{2}-3 a^{2}+7 a^{2}\right)+2 a^{3}+\left(3 a^{5}-4 a^{5}\right)=5 a^{2}+2 a^{3}-a^{5}\]

Solved problems

Simplify        

    \[\frac{-6 a^{-2} b^{4} c^{8}}{24 a^{-5} b^{6} c^{-3}}\]

Solution

Let’s recall our laws of indices from number one to 5. Please identify which laws are applicable in each of the following steps:

    \[\frac{-6 a^{-2} b^{4} c^{8}}{24 a^{-5} b^{6} c^{-3}}=\frac{-6 a^{5} b^{4} c^{8} c^{3}}{24 a^{2} b^{6}} \ldots \ldots \ldots \ldots \ldots \ldots \text { step } 1\]

    \[\frac{-6 a^{5} b^{4} c^{8} c^{3}}{24 a^{2} b^{6}}=\frac{-6 a^{5} b^{4} c^{11}}{24 a^{2} b^{6}} \ldots \ldots \ldots \ldots \ldots \ldots \text { step } 2\]

    \[\frac{-6 a^{5} b^{4} c^{11}}{24 a^{2} b^{6}}=\frac{-6 a^{5} a^{-2} c^{11}}{24 b^{6} b^{-4}}=\frac{-6 a^{3} c^{11}}{24 b^{2}} \ldots \ldots \text { step } 3\]

    \[\frac{-6 a^{3} c^{11}}{24 b^{2}}=\frac{-6\left(a^{3} c^{11}\right)}{6\left(4 b^{2}\right)}=\frac{-a^{3} c^{11}}{4 b^{2}} \ldots \ldots \ldots . . . \text { step } 4\]

Simplify

    \[9 \times 3^{3}\]

Solution

To simplify the given problem, we need to first express both terms to the same base and apply the power rule.

    \[3^{2} \times 3^{3}=3^{3+2}=3^{5}=243\]

Remarks

You can solve any complex problem involving exponents by carefully applying the rules highlighted here. The rules do not change and will never change!

About the Author

This lesson was prepared by Robert O. He holds a Bachelor of Engineering (B.Eng.) degree in Electrical engineering and electronic engineering. He is a career teacher and headed the department of languages and assumed various leadership roles. He writes for Full Potential Learning Academy.