What are distributing exponents?

The distributive property of exponents is a mathematical rule that applies to an exponent that acts on a term that is within parentheses. It says that if there is a single term in the base, such as "3x," and it is raised to a certain power, like "(3x)^6," the exponent applies to all parts of the term, in this case to both the three and the "x."

In the previous example, the distributive property of exponents says that "(3x)^6 = (3^6) (x^6)," and both forms can be used interchangeably.

In another example, "(8 x 4)^3" is the same as saying "(8^3) (4^3)." The problem can be simplified in two ways: (8 x 4)^3 = 32^3 = 32,768 and (8^3) (4^3) = (512) (64) = 32,768. Both ways are equally valid and will always yield the same results.

However, if there are any addition or subtraction terms within the parentheses, there is no longer a single term in the base, so the distributive property of exponents cannot be used in the same way. An example of this is: (6 + 3)^3 ≠ 6^3 + 3^3. This is clearly seen when the inequality is simplified: 9^3 ≠ 216 + 27, because 729 ≠ 243.