You already simplify square roots like $\sqrt{72} = 6\sqrt{2}$ (Adding and Multiplying Radicals). The same idea works for any root, and one more skill finishes the job: clearing roots out of a denominator. It builds on nth Roots.
Idea clave
To simplify $\sqrt[n]{a}$, pull out the largest perfect $n$th power: $\sqrt[3]{54} = \sqrt[3]{27 \cdot 2} = 3\sqrt[3]{2}$. To rationalize $\frac{c}{\sqrt{b}}$, multiply top and bottom by $\sqrt{b}$. To rationalize $\frac{c}{a + \sqrt{b}}$, multiply top and bottom by the conjugate $a - \sqrt{b}$.
Por qué funciona
The product rule $\sqrt[n]{xy} = \sqrt[n]{x}\,\sqrt[n]{y}$ (for nonnegative $x$, $y$) lets you split off the perfect power, whose root is a whole number. Rationalizing multiplies by a form of 1, so the value stays the same. The conjugate works because of the difference of squares: $(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$, which has no root left.
| $n$ | perfect $n$th powers |
|---|---|
| 2 | 4, 9, 16, 25, 36, 49, 64, 81, 100 |
| 3 | 8, 27, 64, 125, 216, 1000 |
| 4 | 16, 81, 256, 625 |
Worked examples
Ejemplo resuelto: A cube root
Simplify $\sqrt[3]{40}$.
- The largest perfect cube factor of 40 is 8: $40 = 8 \cdot 5$.
- $\sqrt[3]{40} = \sqrt[3]{8}\,\sqrt[3]{5} = 2\sqrt[3]{5}$.
Respuesta: $2\sqrt[3]{5}$
Ejemplo resuelto: A single root below
Rationalize $\frac{6}{\sqrt{3}}$.
- Multiply by $\frac{\sqrt{3}}{\sqrt{3}}$: $\frac{6\sqrt{3}}{3}$.
- Simplify: $2\sqrt{3}$.
Respuesta: $2\sqrt{3}$
Ejemplo resuelto: The conjugate
Rationalize $\frac{4}{3 - \sqrt{5}}$.
- Multiply top and bottom by $3 + \sqrt{5}$.
- Bottom: $(3 - \sqrt{5})(3 + \sqrt{5}) = 9 - 5 = 4$.
- Top: $4(3 + \sqrt{5})$. Divide by 4.
Respuesta: $3 + \sqrt{5}$
Error común
Multiplying only the numerator by the conjugate (or by $\sqrt{b}$). That changes the value: you must multiply by $\frac{3 + \sqrt{5}}{3 + \sqrt{5}}$, which is 1. A quick check with a calculator: $\frac{4}{3 - \sqrt{5}} \approx 5.236$ and $3 + \sqrt{5} \approx 5.236$.