How to Divide Exponents With Different Bases

Exponential expressions are common in math, science and finance.
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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.

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