Simplify the Square Root: √(25x⁸/225x⁴) Step-by-Step

Let's solve the problem step by step:

Step 1: Simplify the fraction inside the square root:
$\frac{25x^8}{225x^4}$

Divide both the numerator and the denominator by the greatest common factors. Notice that $25$ and $225$ have a common factor of $25$, and $x^8$ and $x^4$ have a common factor of $x^4$.

The simplification becomes:
$\frac{25 \div 25 \cdot x^{8-4}}{225 \div 25 \cdot x^{4-4}} = \frac{1 \cdot x^4}{9 \cdot 1} = \frac{x^4}{9}$

Step 2: Apply the Quotient Property of Square Roots:
$\sqrt{\frac{x^4}{9}} = \frac{\sqrt{x^4}}{\sqrt{9}}$

Step 3: Simplify the square roots:
Since $\sqrt{x^4} = x^2$ (assuming $x \geq 0$ as square roots imply non-negative results), and $\sqrt{9} = 3$,
$\frac{\sqrt{x^4}}{\sqrt{9}} = \frac{x^2}{3}$

Therefore, the solution to the problem is $\frac{1}{3}x^2$.