Fractional Exponents and Roots
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Main article: Canonical Order of Operations
Fractional exponents and roots are treated in the Canonical Order of Operations as one operation family. A fractional exponent contains both a power and a root: the numerator gives the power, and the denominator gives the root.
Basic rule
x^{\dfrac{m}{n}}=\sqrt[n]{x^m}
In canonical notation, the radical form is rewritten as an index form.
\sqrt[n]{x^m}\;\longrightarrow\;x^{\dfrac{m}{n}}
Examples
| Root expression | Canonical expression | Reading |
|---|---|---|
| \sqrt{x} | x^{\dfrac{1}{2}} | square root of x |
| \sqrt[3]{x} | x^{\dfrac{1}{3}} | cube root of x |
| \sqrt{x^5} | x^{\dfrac{5}{2}} | fifth power under a square root |
| \sqrt[7]{x^3} | x^{\dfrac{3}{7}} | seventh root with a third power |
| \dfrac{1}{\sqrt{x}} | x^{-\dfrac{1}{2}} | reciprocal square root |
Combined power and root
In x^{\dfrac{3}{5}}, the numerator 3 is the power and the denominator 5 is the root. The canonical form keeps both instructions attached to one base.
x^{\dfrac{3}{5}}=\sqrt[5]{x^3}
Nested roots
Nested roots become powers of powers.
\sqrt{\sqrt{x}}=\left(x^{\dfrac{1}{2}}\right)^{\dfrac{1}{2}}=x^{\dfrac{1}{4}}
\sqrt[3]{\sqrt{x}}=\left(x^{\dfrac{1}{2}}\right)^{\dfrac{1}{3}}=x^{\dfrac{1}{6}}
Negative and reciprocal roots
Negative exponents and fractional exponents can be combined.
x^{-\dfrac{1}{2}}=\dfrac{1}{x^{\dfrac{1}{2}}}
x^{-\dfrac{m}{n}}=\dfrac{1}{x^{\dfrac{m}{n}}}
This gives one notation family for powers, roots, reciprocal powers, and reciprocal roots.