8. Understanding Rational Exponents

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Learning Objective:

Use a fractional exponent with numerator 1 to denote a principal root, called a radical expression. Use a more general fractional exponent to denote a principal root raised to a power. Use exponent properties to simplify radical expressions, including rationalizing the denominator of a quotient so that any radical expressions are only in the numerator.

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Roots and radical expressions

If we raise a real number, a, to the power 1/n, where n is a positive integer greater than 1, we obtain the principal n-th root of a: a¹/ⁿ = ⁿ√a. An expression containing such a root is called a radical expression.

If a > 0, then ⁿ√a is the positive real number b such that bⁿ = a. For example, ³√8 = 2 because 2³ = 8 and ²√16 = √16 = 4 because 4² = 16. Note that -4 is also a square root of 16 because (-4)² = 16, but it is not the principal square root.
If a < 0 and n is odd, then ⁿ√a is the negative real number b such that bⁿ = a. For example, ³√-8 = -2 because (-2)³ = -8.
If a < 0 and n is even, then ⁿ√a is not a real number. For example, ²√-16 is not a real number because there is no real number that we can square to equal -16.
If a = 0, then ⁿ√a = 0.

More Examples

$\sqrt[3]{27} = 3 \text{ because } 3^3 = 27.$

$\sqrt{36} = 6 \text{ because } 6^2 = 36 \text{ (and -6 is also a square root of 36).}$

$\sqrt[5]{-32} = -2 \text{ because } (-2)^5 = -32.$

$\sqrt[4]{\frac{1}{81}} = \frac{1}{3} \text{ because } \left(\frac{1}{3}\right)^4 = \frac{1}{3^4} = \frac{1}{81} \text{ (and } -\frac{1}{3} \text{ is also a fourth root of } \frac{1}{81}).$

Why it makes sense to define roots this way

It makes sense because (a¹^/ⁿ)ⁿ = (ⁿ√a)ⁿ = a. For example, (³√8)³ = 2³ = 8.

More general rational exponents

If we raise a real number, a, to the power m/n, where m is an integer and n is a positive integer greater than 1, we obtain the principal n-th root of a raised to the power m (in either order): aᵐ/ⁿ = (ⁿ√a)ᵐ = ⁿ√(aᵐ). For example, 8²^/³ = (³√8)² = 2² = 4 or 8²^/³ = ³√8² = ³√64 = 4.

More Examples

$4^{3/2} = (\sqrt{4})^3 = 2^3 = 8 \text{ or } 4^{3/2} = \sqrt{4^3} = \sqrt{64} = 8.$

$(-8)^{2/3} = (\sqrt[3]{-8})^2 = (-2)^2 = 4 \text{ or } (-8)^{2/3} = \sqrt[3]{(-8)^2} = \sqrt[3]{64} = 4.$

$81^{-3/4} = \frac{1}{81^{3/4}} = \frac{1}{(\sqrt[4]{81})^3} = \frac{1}{3^3} = \frac{1}{27}. \text{ (The other way is harder.)}$

$\left(\frac{9}{16}\right)^{3/2} = \frac{9^{3/2}}{16^{3/2}} = \frac{(\sqrt{9})^3}{(\sqrt{16})^3} = \frac{3^3}{4^3} = \frac{27}{64}. \text{ (The other way is harder.)}$

Rationalizing the denominator

Sometimes, we want any radical expressions to only be in the numerator. We can do this by rationalizing the denominator by multiplying by the appropriate fraction equivalent to 1. For example:

$\frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

$\frac{1}{\sqrt[3]{2}} = \frac{1}{\sqrt[3]{2}} \cdot \frac{\sqrt[3]{2^2}}{\sqrt[3]{2^2}} = \frac{\sqrt[3]{2^2}}{2} = \frac{\sqrt[3]{4}}{2}$

$\left(\frac{x}{2y}\right)^{1/3} = \frac{x^{1/3}}{(2y)^{1/3}} \cdot \frac{(2y)^{2/3}}{(2y)^{2/3}} = \frac{(4xy^2)^{1/3}}{2y} = \frac{\sqrt[3]{4xy^2}}{2y}$

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8. Understanding Rational Exponents

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