Question

simplify (sqrt{x^3})

Explanation:

Step1: Rewrite the square root as an exponent

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

Step2: Apply the power - of - a - power rule

The power - of - a - power rule states that $(a^{m})^{n}=a^{m\times n}$. For $(x^{3})^{\frac{1}{2}}$, we have $m = 3$ and $n=\frac{1}{2}$, so $(x^{3})^{\frac{1}{2}}=x^{3\times\frac{1}{2}}=x^{\frac{3}{2}}$.

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

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

Answer:

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