Because fractional exponents were defined to keep the exponent rules true, you can now use those rules on roots. Rewriting roots as powers often turns a messy radical problem into simple fraction arithmetic.
| Rule | Example with fractions |
|---|---|
| $x^a \cdot x^b = x^{a+b}$ | $x^{1/2} \cdot x^{1/3} = x^{5/6}$ |
| $\frac{x^a}{x^b} = x^{a-b}$ | $\frac{x^{3/4}}{x^{1/4}} = x^{1/2}$ |
| $(x^a)^b = x^{ab}$ | $\left(x^{2/3}\right)^{3/4} = x^{1/2}$ |
| $(xy)^a = x^a y^a$ | $\left(8x^6\right)^{1/3} = 2x^2$ |
Por qué funciona
Why add exponents when multiplying? $x^3 \cdot x^2$ is three factors of $x$ times two more, five in all. With fractions, think of $x^{1/6}$ as a basic piece: $x^{1/2} = \left(x^{1/6}\right)^3$ and $x^{1/3} = \left(x^{1/6}\right)^2$, so their product is $\left(x^{1/6}\right)^5 = x^{5/6}$. Adding $\frac{1}{2} + \frac{1}{3}$ is counting those pieces. A power of a power repeats a whole block of factors, which is why those exponents multiply.
Idea clave
Same base: multiply means add the exponents, divide means subtract them. A power of a power multiplies the exponents. A power of a product applies to every factor.
Worked examples
Ejemplo resuelto: Roots of different index
Write $\sqrt{x} \cdot \sqrt[3]{x}$ as one power of $x$.
- Rewrite: $x^{1/2} \cdot x^{1/3}$.
- Add the exponents: $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$.
Respuesta: $x^{5/6}$, which is $\sqrt[6]{x^5}$.
Ejemplo resuelto: Power of a product
Simplify $\left(16x^8\right)^{3/4}$.
- Apply the power to each factor: $16^{3/4} \cdot \left(x^8\right)^{3/4}$.
- $16^{3/4} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8$.
- $\left(x^8\right)^{3/4} = x^{8 \cdot 3/4} = x^6$.
Respuesta: $8x^6$
Ejemplo resuelto: A quotient
Simplify $\dfrac{x^{5/3}}{x^{1/2}}$.
- Subtract the exponents: $\frac{5}{3} - \frac{1}{2} = \frac{10}{6} - \frac{3}{6} = \frac{7}{6}$.
Respuesta: $x^{7/6}$
Error común
Adding the exponents when raising a power to a power, writing $\left(x^{2/3}\right)^{3/4} = x^{2/3 + 3/4}$. Adding is for multiplying two powers; a power of a power multiplies: $\frac{2}{3} \cdot \frac{3}{4} = \frac{1}{2}$. Check with whole numbers you trust: $(x^2)^3 = x^2 \cdot x^2 \cdot x^2 = x^6$, not $x^5$.