Solve (sqrt(5) - sqrt(2))/(sqrt(5) + sqrt(2)) = | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+\sqrt{2}}=$$

Evaluate

$\frac{7-2\sqrt{10}}{3}\approx 0.225148227$

Solution Steps

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

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

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

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

Square $\sqrt{5}$. Square $\sqrt{2}$.

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

Subtract $2$ from $5$ to get $3$.

$$\frac{\left(\sqrt{5}-\sqrt{2}\right)\left(\sqrt{5}-\sqrt{2}\right)}{3}$$

Multiply $\sqrt{5}-\sqrt{2}$ and $\sqrt{5}-\sqrt{2}$ to get $\left(\sqrt{5}-\sqrt{2}\right)^{2}$.

$$\frac{\left(\sqrt{5}-\sqrt{2}\right)^{2}}{3}$$

Use binomial theorem $\left(a-b\right)^{2}=a^{2}-2ab+b^{2}$ to expand $\left(\sqrt{5}-\sqrt{2}\right)^{2}$.

$$\frac{\left(\sqrt{5}\right)^{2}-2\sqrt{5}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{3}$$

The square of $\sqrt{5}$ is $5$.

$$\frac{5-2\sqrt{5}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{3}$$

To multiply $\sqrt{5}$ and $\sqrt{2}$, multiply the numbers under the square root.

$$\frac{5-2\sqrt{10}+\left(\sqrt{2}\right)^{2}}{3}$$

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

$$\frac{5-2\sqrt{10}+2}{3}$$

Add $5$ and $2$ to get $7$.

$$\frac{7-2\sqrt{10}}{3}$$