Question

simplify.
\sqrt{4x^{13}}
assume that the variable represents a positive real number.

Explanation:

Step1: Simplify the square root of the constant term

We know that $\sqrt{4} = 2$ because $2^2 = 4$.

Step2: Simplify the square root of the variable term

For the variable part $x^{13}$, we can use the property of exponents $\sqrt{x^n}=x^{\frac{n}{2}}$ (when $x\gt0$). We can rewrite $x^{13}$ as $x^{12}\cdot x$. Then $\sqrt{x^{13}}=\sqrt{x^{12}\cdot x}=\sqrt{x^{12}}\cdot\sqrt{x}$. Since $\sqrt{x^{12}} = x^6$ (because $(x^6)^2=x^{12}$), we have $\sqrt{x^{13}}=x^6\sqrt{x}$.

Step3: Combine the simplified constant and variable terms

Now, we combine the results from Step 1 and Step 2. $\sqrt{4x^{13}}=\sqrt{4}\cdot\sqrt{x^{13}} = 2\cdot x^6\sqrt{x}=2x^6\sqrt{x}$.

Answer:

$2x^{6}\sqrt{x}$