Solve (1)/(1+sqrt(2))+(1)/(sqrt(2)\timessqrt(3))+(1)/(sqrt(3+sqrt(4))) | 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

$$\frac{ 1 }{ 1+ \sqrt{ 2 } } + \frac{ 1 }{ \sqrt{ 2 } \times \sqrt{ 3 } } + \frac{ 1 }{ \sqrt{ 3+ \sqrt{ 4 } } }$$

Evaluate

$\frac{\sqrt{5}}{5}+\frac{\sqrt{6}}{6}+\sqrt{2}-1\approx 1.269675448$

Short Solution Steps

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

$$\frac{1-\sqrt{2}}{\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)}+\frac{1}{\sqrt{2}\sqrt{3}}+\frac{1}{\sqrt{3+\sqrt{4}}}$$

Consider $\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{1-\sqrt{2}}{1^{2}-\left(\sqrt{2}\right)^{2}}+\frac{1}{\sqrt{2}\sqrt{3}}+\frac{1}{\sqrt{3+\sqrt{4}}}$$

Square $1$. Square $\sqrt{2}$.

$$\frac{1-\sqrt{2}}{1-2}+\frac{1}{\sqrt{2}\sqrt{3}}+\frac{1}{\sqrt{3+\sqrt{4}}}$$

Subtract $2$ from $1$ to get $-1$.

$$\frac{1-\sqrt{2}}{-1}+\frac{1}{\sqrt{2}\sqrt{3}}+\frac{1}{\sqrt{3+\sqrt{4}}}$$

Anything divided by -1 gives its opposite. To find the opposite of $1-\sqrt{2}$, find the opposite of each term.

$$-1-\left(-\sqrt{2}\right)+\frac{1}{\sqrt{2}\sqrt{3}}+\frac{1}{\sqrt{3+\sqrt{4}}}$$

The opposite of $-\sqrt{2}$ is $\sqrt{2}$.

$$-1+\sqrt{2}+\frac{1}{\sqrt{2}\sqrt{3}}+\frac{1}{\sqrt{3+\sqrt{4}}}$$

To multiply $\sqrt{2}$ and $\sqrt{3}$, multiply the numbers under the square root.

$$-1+\sqrt{2}+\frac{1}{\sqrt{6}}+\frac{1}{\sqrt{3+\sqrt{4}}}$$

Rationalize the denominator of $\frac{1}{\sqrt{6}}$ by multiplying numerator and denominator by $\sqrt{6}$.

$$-1+\sqrt{2}+\frac{\sqrt{6}}{\left(\sqrt{6}\right)^{2}}+\frac{1}{\sqrt{3+\sqrt{4}}}$$

The square of $\sqrt{6}$ is $6$.

$$-1+\sqrt{2}+\frac{\sqrt{6}}{6}+\frac{1}{\sqrt{3+\sqrt{4}}}$$

Calculate the square root of $4$ and get $2$.

$$-1+\sqrt{2}+\frac{\sqrt{6}}{6}+\frac{1}{\sqrt{3+2}}$$

Add $3$ and $2$ to get $5$.

$$-1+\sqrt{2}+\frac{\sqrt{6}}{6}+\frac{1}{\sqrt{5}}$$

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

$$-1+\sqrt{2}+\frac{\sqrt{6}}{6}+\frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

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

$$-1+\sqrt{2}+\frac{\sqrt{6}}{6}+\frac{\sqrt{5}}{5}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply $-1+\sqrt{2}$ times $\frac{6}{6}$.

$$\frac{6\left(-1+\sqrt{2}\right)}{6}+\frac{\sqrt{6}}{6}+\frac{\sqrt{5}}{5}$$

Since $\frac{6\left(-1+\sqrt{2}\right)}{6}$ and $\frac{\sqrt{6}}{6}$ have the same denominator, add them by adding their numerators.

$$\frac{6\left(-1+\sqrt{2}\right)+\sqrt{6}}{6}+\frac{\sqrt{5}}{5}$$

Do the multiplications in $6\left(-1+\sqrt{2}\right)+\sqrt{6}$.

$$\frac{-6+6\sqrt{2}+\sqrt{6}}{6}+\frac{\sqrt{5}}{5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $6$ and $5$ is $30$. Multiply $\frac{-6+6\sqrt{2}+\sqrt{6}}{6}$ times $\frac{5}{5}$. Multiply $\frac{\sqrt{5}}{5}$ times $\frac{6}{6}$.

$$\frac{5\left(-6+6\sqrt{2}+\sqrt{6}\right)}{30}+\frac{6\sqrt{5}}{30}$$

Since $\frac{5\left(-6+6\sqrt{2}+\sqrt{6}\right)}{30}$ and $\frac{6\sqrt{5}}{30}$ have the same denominator, add them by adding their numerators.

$$\frac{5\left(-6+6\sqrt{2}+\sqrt{6}\right)+6\sqrt{5}}{30}$$

Do the multiplications in $5\left(-6+6\sqrt{2}+\sqrt{6}\right)+6\sqrt{5}$.

$$\frac{-30+30\sqrt{2}+5\sqrt{6}+6\sqrt{5}}{30}$$