Solve for X: Square Root Equation sqrt(2)·sqrt(4)/sqrt(16) = x/sqrt(8)

Let's solve the equation step by step:

• Step 1: Simplify the left side of the equation
  We are given $\frac{\sqrt{2}\cdot\sqrt{4}}{\sqrt{16}}$. Start by simplifying each square root:
  • $\sqrt{2}$ remains as it is.
  • $\sqrt{4} = 2$ because $4 = 2^2$.
  • $\sqrt{16} = 4$ because $16 = 4^2$.
  Substitute these into the initial expression: $\frac{\sqrt{2} \cdot 2}{4} = \frac{2\sqrt{2}}{4}$.
• Step 2: Simplify the expression
  The expression $\frac{2\sqrt{2}}{4}$ simplifies to $\frac{\sqrt{2}}{2}$.
• Step 3: Equate to the right side
  Set this equal to the right side: $\frac{\sqrt{2}}{2} = \frac{x}{\sqrt{8}}$.
• Step 4: Cross-multiply to solve for $x$
  Cross-multiplying gives $x \cdot 2 = \sqrt{2} \cdot \sqrt{8}$. Simplify the right side:
  • $\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$
  • So, $\sqrt{2} \cdot \sqrt{8} = \sqrt{2} \cdot 2\sqrt{2} = 2 \cdot (\sqrt{2} \cdot \sqrt{2}) = 2 \cdot 2 = 4$
  • The equation becomes $2x = 4$.
  • Solving for $x$, divide both sides by 2: $x = \frac{4}{2} = 2$.

Therefore, the solution to the problem is $x = 2$.