How to Divide Exponents With Different Bases

Exponential expressions are common in math, science and finance.
••• marekuliasz/iStock/Getty Images

An exponent is a number, usually written as a superscript or after the caret symbol ^, that indicates repeated multiplication. The number being multiplied is called the base. If b is the base and n is the exponent, we say “b to the power of n,” shown as b^n, which means b * b * b * b ...* b n times. For example “4 to the power of 3” means 4^3 = 4 * 4 * 4 = 64. There are rules for doing operations on exponential expressions. Dividing exponential expressions with different bases is allowed but poses unique problems when it comes to simplification, which can only sometimes be done.

Different Bases and Same Exponent

In this case, you can group the two bases into a quotient and apply the exponent. For example, 5^3 / 7^3 = (5 / 7)^3. With variables, b^3 / c^3 = (b * b * b) / (c * c * c) = (b / c) * (b / c) * (b / c) = (b / c)^3. In general, b^n / c^n = (b / c)^n.

Different Bases and Different Exponents

The expression b^4 / a^2 is equivalent to (b * b * b * b) / (a * a). Nothing cancels here, but you can transform the expression by grouping by exponents. For example, b^4 / a^2 = (b / a)^2 * b^2, or (b^2 / a)^2. In some cases a transformation creates an expression that is simpler in the sense that it eliminates common factors and reduces the magnitude of the numbers in the expression. For example: 120^3 / 40^5 = (120 / 40)^3 / 4^2 = 3^3 / 4^2. Unfortunately, that is as “simple” as you can get without evaluating the number.

Order of Operations

Powers are higher in precedence than multiplication and division. So to evaluate the expression 3^3 / 4^2, you do the exponentiation first and the division second: 3^3 / 4^2 = 9 / 16 = 0.5265.

Related Articles

How to Find Missing Exponents
How to Get Rid of Cubed Power
How to Add & Multiply Exponents
How to Simplify Monomials
How to Factor Polynomials With 4 Terms
How to Simplify Exponents
How to Cancel a Natural Log
How to Add & Subtract Radical Expressions With Fractions
How to Convert Fractions to Exponential Notation
How to Do Exponents Outside of the Parenthesis
Algebra Rules for Beginners
How to Factor With Negative Fractional Exponents
Calculating Logarithms
How to Add & Subtract Fractions With Monomials
How to Get Rid of Logarithms
Negative Exponents: Rules for Multiplying & Dividing
Steps in Learning How to Do Long Division With Bases...
How to Calculate Exponents
How to Divide Negative Numbers
How Can I Remember Math Properties?

Dont Go!

We Have More Great Sciencing Articles!