1. Factor the equation (x + 2)(x + 3) = 0. 2. Set each factor equal to zero: x + 2 = 0 and x + 3 = 0. 3. Solve for x: x = -2 and x = -3.

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Question

$$\frac { 1 } { \sqrt { 2 } } \times \frac { 1 } { \sqrt { 6 } }$$

Evaluate

$\frac{\sqrt{3}}{6}\approx 0.288675135$

Solution Steps

Rationalize the denominator of $\frac{1}{\sqrt{2}}$ by multiplying numerator and denominator by $\sqrt{2}$.

$$\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\times \frac{1}{\sqrt{6}}$$

The square of $\sqrt{2}$ is $2$.

$$\frac{\sqrt{2}}{2}\times \frac{1}{\sqrt{6}}$$

Rationalize the denominator of $\frac{1}{\sqrt{6}}$ by multiplying numerator and denominator by $\sqrt{6}$.

$$\frac{\sqrt{2}}{2}\times \frac{\sqrt{6}}{\left(\sqrt{6}\right)^{2}}$$

The square of $\sqrt{6}$ is $6$.

$$\frac{\sqrt{2}}{2}\times \frac{\sqrt{6}}{6}$$

Multiply $\frac{\sqrt{2}}{2}$ times $\frac{\sqrt{6}}{6}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{2}\sqrt{6}}{2\times 6}$$

Factor $6=2\times 3$. Rewrite the square root of the product $\sqrt{2\times 3}$ as the product of square roots $\sqrt{2}\sqrt{3}$.

$$\frac{\sqrt{2}\sqrt{2}\sqrt{3}}{2\times 6}$$

Multiply $\sqrt{2}$ and $\sqrt{2}$ to get $2$.

$$\frac{2\sqrt{3}}{2\times 6}$$

Multiply $2$ and $6$ to get $12$.

$$\frac{2\sqrt{3}}{12}$$

Divide $2\sqrt{3}$ by $12$ to get $\frac{1}{6}\sqrt{3}$.

$$\frac{1}{6}\sqrt{3}$$

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