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Rational Exponent Rule
A rational exponent of (p/q) on a base of x would be written x^(p/q). This can be rewritten as a radical with "q" as the index number, "x" as the number within the radical and "p" as the exponent applied to the "x." For example, x^(1/2) would equal √(x^1). This would also be equivalent to (√x)^1.
Product and Quotient Rules
The product rule of exponents states that x^a * x^b = x^(a + b). Note that the bases must be the same for this rule to work. A rational exponent example: x^(2/3) * x^(1/3) = x^ (2 + 1 / 3) = x^(3/3) = x^1 = x.
The quotient rule of exponents states that (x^a) / (x^b) = x^(a - b). A rational exponent example: (x^(2/5)) / (x^(1/3)) = x^((2/5) - (1/3)). Convert the fractions to the lowest common denominator: x^((6/15) - (5/15)) = x^(1/15).
Power Rules
The power rule for exponents states that (x^a)^b = x^(a * b). A rational exponent example: (x^(3/5))^(2/3) = x^((3/5) * (2/3)) = x^(6/15). Simplify the fraction: x^(2/5).
The other two power rules apply to problems with different bases. The products to power rule states that (xy)^a = x^a * y^a. For example, (xy)^(1/4) = x^(1/4) * y^(1/4). The quotient to power rule states that (x/y)^a = (x^a) / (y^a). For example, (x/y)^(2/3) = (x^(2/3)) / (y^(2/3)).
Negative Exponent Rule
When applying the negative exponent rule, it's very important to pay attention to signs. The rule states that x^(-a) = 1 / x^a. The rule also says that 1/x^(-a) becomes x^a. For example, x^(-3/4) = 1/ x^(3/4). Or 1 / x^(-2/3) = x^(2/3).