Solve sqrt(5)+2+(1)/(sqrt(5)+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

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

Evaluate

$2\sqrt{5}\approx 4.472135955$

Solution Steps

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

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

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

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

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

$$\sqrt{5}+2+\frac{\sqrt{5}-2}{5-4}$$

Subtract $4$ from $5$ to get $1$.

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

Anything divided by one gives itself.

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

Combine $\sqrt{5}$ and $\sqrt{5}$ to get $2\sqrt{5}$.

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

Subtract $2$ from $2$ to get $0$.

$$2\sqrt{5}$$