How to solve 1/(8 + 5sqrt(2))

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/(8 + 5sqrt(2)) | Equatix

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Question

$$\frac{1}{8+5\sqrt{2}}$$

Evaluate

$-\frac{5\sqrt{2}}{14}+\frac{4}{7}\approx 0.066352299$

Solution Steps

Rationalize the denominator of $\frac{1}{8+5\sqrt{2}}$ by multiplying numerator and denominator by $8-5\sqrt{2}$.

$$\frac{8-5\sqrt{2}}{\left(8+5\sqrt{2}\right)\left(8-5\sqrt{2}\right)}$$

Consider $\left(8+5\sqrt{2}\right)\left(8-5\sqrt{2}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{8-5\sqrt{2}}{8^{2}-\left(5\sqrt{2}\right)^{2}}$$

Calculate $8$ to the power of $2$ and get $64$.

$$\frac{8-5\sqrt{2}}{64-\left(5\sqrt{2}\right)^{2}}$$

Expand $\left(5\sqrt{2}\right)^{2}$.

$$\frac{8-5\sqrt{2}}{64-5^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate $5$ to the power of $2$ and get $25$.

$$\frac{8-5\sqrt{2}}{64-25\left(\sqrt{2}\right)^{2}}$$

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

$$\frac{8-5\sqrt{2}}{64-25\times 2}$$

Multiply $25$ and $2$ to get $50$.

$$\frac{8-5\sqrt{2}}{64-50}$$

Subtract $50$ from $64$ to get $14$.

$$\frac{8-5\sqrt{2}}{14}$$

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