Question

cated operation. assume all variables are positive. \\(\sqrt{3}\\) \\(\sqrt{9} \cdot \sqrt{3} = \\) 11.) \\(\sqrt{(\sqrt4{16x} - \sqrt4{x}\\)

Explanation:

Step1: Simplify the 4th root term

First, rewrite $\sqrt[4]{16x}$:
$\sqrt[4]{16x} = \sqrt[4]{16} \cdot \sqrt[4]{x} = 2\sqrt[4]{x}$

Step2: Subtract the like radical terms

Substitute back into the expression inside the square root:
$\sqrt{(2\sqrt[4]{x} - \sqrt[4]{x})} = \sqrt{(\sqrt[4]{x})}$

Step3: Rewrite roots as exponents

Convert the nested roots to fractional exponents to simplify:
$\sqrt{\sqrt[4]{x}} = (x^{\frac{1}{4}})^{\frac{1}{2}}$

Step4: Apply exponent power rule

Use the rule $(a^m)^n = a^{m \cdot n}$:
$(x^{\frac{1}{4}})^{\frac{1}{2}} = x^{\frac{1}{4} \cdot \frac{1}{2}} = x^{\frac{1}{8}}$

Step5: Convert back to radical form (optional)

$x^{\frac{1}{8}} = \sqrt[8]{x}$

Answer:

$\sqrt[8]{x}$ (or $x^{\frac{1}{8}}$)