Writing Radicals with Fractional Exponents

From radical signs to fractional exponents

Reverse the rule \sqrt[n]{x^a} = x^{a/n}, and give each factor of the radicand its own exponent: \sqrt[3]{8x^6y} = 8^{1/3}x^{6/3}y^{1/3} = 2x^2y^{1/3}.

Simplify the results:

work out numbers that have exact roots (16^{1/2} = 4, 8^{2/3} = 4);
reduce each exponent (x^{6/3} = x^2);
multiply powers of the same base by adding exponents: \sqrt x \cdot \sqrt[4]{x} = x^{1/2}x^{1/4} = x^{3/4}.

A radicand that is a sum, like (a + b)^3 under a square root, keeps its parentheses: \sqrt{(a + b)^3} = (a + b)^{3/2}.

Typing answers: fractional exponents go in parentheses: 2x^2*y^(1/3), (a + b)^(3/2). Don't use radical signs in these answers, and give each letter its own exponent: a^(1/2)x^2, not (ax^4)^(1/2). A number that has no exact root keeps a fractional exponent: 3*2^(1/2) or 2^(4/3).

Exercises

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How to type math

Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.

Write with fractional exponents and simplify results:

\sqrt[2]{a^3}

\sqrt[2]{ax^4}

3\sqrt{2x^5}

\sqrt{9x}

5\sqrt{16ax^2}

\sqrt[3]{a^2}

\sqrt[3]{ax^4}

2\sqrt[3]{2x^2}

3\sqrt[3]{8x^4}

4\sqrt[3]{27ax^3}

2\sqrt[4]{a^2x^8}

4\sqrt[4]{16x}

7\sqrt[5]{rs^7}

5a\sqrt[5]{32m^2}

12x^2\sqrt[3]{ax^3}

c\sqrt[2]{(de)^3}

uv\sqrt[2]{(u + v)^5}

3\sqrt[2]{a^3}\cdot\sqrt[3]{x^2}

2a\sqrt[5]{ax^2}\cdot\sqrt[3]{2m}

\sqrt[n]{x^a}\cdot\sqrt[n]{y^b}

\sqrt[n]{x^a}\cdot\sqrt[n]{x^{3a}}