Solve (iii ) 1/(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

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

Evaluate

$\frac{\sqrt{5}-\sqrt{2}}{3}\approx 0.273951472$

Solution Steps

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

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

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

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

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

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

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

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