Question

simplify (sqrt{x^3})

Explanation:

Step1: Rewrite the radical as exponent

Recall that $\sqrt{a}=a^{\frac{1}{2}}$, so $\sqrt{x^{3}}=(x^{3})^{\frac{1}{2}}$.

Step2: Apply exponent rule $(a^{m})^{n}=a^{mn}$

Using the rule, we have $(x^{3})^{\frac{1}{2}} = x^{3\times\frac{1}{2}}=x^{\frac{3}{2}}$.

Step3: Rewrite the exponent as a mixed number (optional, but common for simplification)

We can write $x^{\frac{3}{2}}$ as $x^{1+\frac{1}{2}}$. Then, using the rule $a^{m + n}=a^{m}\times a^{n}$, we get $x^{1}\times x^{\frac{1}{2}}=x\sqrt{x}$ (since $x^{\frac{1}{2}}=\sqrt{x}$).

Answer:

$x\sqrt{x}$ (or $x^{\frac{3}{2}}$)