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

$4$

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.

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

To find the opposite of $\sqrt{5}-2$, find the opposite of each term.

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

The opposite of $-2$ is $2$.

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

Combine $\sqrt{5}$ and $-\sqrt{5}$ to get $0$.

$$2+2$$

Add $2$ and $2$ to get $4$.

$$4$$

Factor

$2^{2}$