If $\sqrt{2}:(1 + \sqrt{3}) :: \sqrt{6}: x$, then $x$ is equal to:

The symbol "::" represents a proportion. When we write $\sqrt{2}:(1 + \sqrt{3}) :: \sqrt{6}: x$, it means:

$\frac{\sqrt{2}}{1 + \sqrt{3}} = \frac{\sqrt{6}}{x}$

Think of it as: "√2 is to (1+√3) as √6 is to x"

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Cross-multiply:

$\frac{\sqrt{2}}{1 + \sqrt{3}} = \frac{\sqrt{6}}{x}$

$\sqrt{2} \cdot x = \sqrt{6} \cdot (1 + \sqrt{3})$

$x \cdot \sqrt{2} = \sqrt{6} + \sqrt{6} \cdot \sqrt{3}$

When you multiply square roots, you multiply what's inside: $\sqrt{6} \times \sqrt{3} = \sqrt{6 \times 3} = \sqrt{18}$

Simplify $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$

$x \cdot \sqrt{2} = \sqrt{6} + 3\sqrt{2}$

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Divide both sides by $\sqrt{2}$:

$x = \frac{\sqrt{6} + 3\sqrt{2}}{\sqrt{2}}$

Split into two fractions:

$x = \frac{\sqrt{6}}{\sqrt{2}} + \frac{3\sqrt{2}}{\sqrt{2}}$

$x = \frac{\sqrt{6}}{\sqrt{2}} + 3$

When dividing square roots, divide what's inside: $\sqrt{6} \div \sqrt{2} = \sqrt{6 \div 2} = \sqrt{3}$

$x = \sqrt{3} + 3$