How to solve 1/(sqrt(2) + 3)

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.

Solve 1/(sqrt(2) + 3) | Equatix

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Question

$$\frac{1}{\sqrt{2}+3}$$

Evaluate

$\frac{3-\sqrt{2}}{7}\approx 0.22654092$

Solution Steps

Rationalize the denominator of $\frac{1}{\sqrt{2}+3}$ by multiplying numerator and denominator by $\sqrt{2}-3$.

$$\frac{\sqrt{2}-3}{\left(\sqrt{2}+3\right)\left(\sqrt{2}-3\right)}$$

Consider $\left(\sqrt{2}+3\right)\left(\sqrt{2}-3\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{\sqrt{2}-3}{\left(\sqrt{2}\right)^{2}-3^{2}}$$

Square $\sqrt{2}$. Square $3$.

$$\frac{\sqrt{2}-3}{2-9}$$

Subtract $9$ from $2$ to get $-7$.

$$\frac{\sqrt{2}-3}{-7}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{2}+3}{7}$$

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