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The Rules of Simplifying Integer Exponents

An exponent denotes how many times a number, called the base, should be multiplied by itself. For example, 4^3 equals to 4 * 4 * 4. When an exponent is applied to a variable, it usually can't be solved but can be simplified using one of the rules for integer exponents.

Product Rule for Exponents
- The product rule for exponents states that x^a * x^b = x^(a + b). In other words, if the bases in a multiplication are the same and the exponents differ, the result would be the base raised to the addition of the exponents. For example, x^3 * x^5 = x^(3 + 5) = x^8.
Quotient Rule for Exponents
- The quotient rule for exponents states that (x^a) / (x^b) = x^(a - b). This means that when there is a division problem with the same base in the numerator and denominator, but differing exponents, the result is the base raised to the subtraction of the lower exponent from the upper exponent. For example, (x^10) / (x^6) = x^(10 - 6) = x^4.
Power Rule for Exponents
- The power rule for exponents states that (x^a)^b = x^(a * b). This means that a base raised to an exponent within a parenthesis, then raised by an exterior exponent, will become the base raised to the two exponents multiplied. For example, (x^2)^3 = x^(2 * 3) = x^6.
Differing Bases
- There are two exponential rules for when there are differing bases.
  
  The products to powers rule for exponents states that (xy)^a = x^a * y^a. This means that an exterior exponent, outside a parenthesis, should be distributed to each term within. For example, (xy)^3 becomes (x^3) * (y^3).
  
  The quotients to powers rule for exponents states that (x/ y)^a = (x^a) / (y^a). Again, this shows that the exterior exponent should be distributed to each term within with the algebraic operation maintained. For example, (x / y)^8 = (x^8) / (y^8).

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