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Rational Exponent
Positive Rational Exponent:We know that 2^{3} = 8. It can also be expressed as 8^{1/3 } = 2.In general if x and yare nonzero rational numbers and m is a positive integer such that x^{m} = y then we can also express it as y^{1/m} = x but we can write y^{1/m} = √(m&y) and is called m^{th} root of y. For example, y^{1/2} = √(2&y), y^{1/3} = ∛y, y^{1/4} = ∜y, etc. If x is a positive rational number then for a positive ration exponent p/q we have x_{p/q }can be defined in two equivalent form. x^{p/q} = (x^{p})^{1/q} = √(q&x)p is read as q^{th} root of x^{p} x^{p/q} = (x^{1/q})^{p} = (√(q&x))p is read as p^{th} power of q^{th} root of x For example: 1. Find (125)^{2/3}. Solution: (125)^{2/3} 125 can be expressed as 5 × 5 × 5 = 5^{3} So, we have (125)^{2/3} = (5^{3})^{2/3} = 5^{3 × 2/3} = 5^{2} = 25 2. Find (8/27)^{4/3} Solution: (8/27)^{4/3} 8 = 2^{3} and 27 = 3^{3} So, we have (8/27)^{4/3} = (2^{3}/3^{3})^{4/3} = [(2/3) ^{3}]^{4/3} = (2/3)^{3 × 4/3} = (2/3) ^{4} = 2/3 × 2/3 × 2/3 × 2/3 = 16/81 3. Find 9^{1/2} Solution: 9^{1/2} = √(2&9) = [(3)^{2}]^{1/2} = (3)^{2 × 1/2} = 3 4. Find 125^{1/3} Solution: 125^{1/3} = ∛125 = [(5) ^{3}]^{1/3} = (5) ^{3 × 1/3} = 5 Negative Rational Exponent:We already learnt that if x is a nonzero rational number and m is any positive integer then x^{m} = 1/x^{m} = (1/x)^{m}, i.e., x^{m} is the reciprocal of x^{m}.Same rule exists of rational exponents. If p/q is a positive rational number and x > 0 is a rational number Then x^{p/q} = 1/x^{p/q} = (1/x) ^{p/q}, i.e., x^{p/q} is the reciprocal of x^{p/q} If x = a/b, then (a/b)^{p/q} = (b/a)^{p/q} For example: 1. Find 9^{1/2} Solution: 9^{1/2} = 1/9^{1/2} = (1/9)^{1/2} = [(1/3)^{2}]^{1/2} = (1/3)^{2 × 1/2} = 1/3 2. Find (27/125)^{4/3} Solution: (27/125)^{4/3} = (125/27)^{4/3} = (5^{3}/3^{3})^{4/3} = [(5/3) ^{3}]^{4/3} = (5/3)^{3 × 4/3} = (5/3)^{4} = (5 × 5 × 5 × 5)/(3 × 3 × 3 × 3) = 625/81 Exponents Exponents  Worksheets


