Solve lify (1 + sqrt(2))/(sqrt(5) + sqrt(3)) + (1 - sqrt(2))/(sqrt(5) - sqrt(3)) | 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

$$\frac { 1 + \sqrt { 2 } } { \sqrt { 5 } + \sqrt { 3 } } + \frac { 1 - \sqrt { 2 } } { \sqrt { 5 } - \sqrt { 3 } }$$

Evaluate

$\sqrt{5}-\sqrt{6}\approx -0.213421765$

Short Solution Steps

Rationalize the denominator of $\frac{1+\sqrt{2}}{\sqrt{5}+\sqrt{3}}$ by multiplying numerator and denominator by $\sqrt{5}-\sqrt{3}$.

$$\frac{\left(1+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}+\frac{1-\sqrt{2}}{\sqrt{5}-\sqrt{3}}$$

Consider $\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{\left(1+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}+\frac{1-\sqrt{2}}{\sqrt{5}-\sqrt{3}}$$

Square $\sqrt{5}$. Square $\sqrt{3}$.

$$\frac{\left(1+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{5-3}+\frac{1-\sqrt{2}}{\sqrt{5}-\sqrt{3}}$$

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

$$\frac{\left(1+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2}+\frac{1-\sqrt{2}}{\sqrt{5}-\sqrt{3}}$$

Rationalize the denominator of $\frac{1-\sqrt{2}}{\sqrt{5}-\sqrt{3}}$ by multiplying numerator and denominator by $\sqrt{5}+\sqrt{3}$.

$$\frac{\left(1+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2}+\frac{\left(1-\sqrt{2}\right)\left(\sqrt{5}+\sqrt{3}\right)}{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}$$

Consider $\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{\left(1+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2}+\frac{\left(1-\sqrt{2}\right)\left(\sqrt{5}+\sqrt{3}\right)}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

Square $\sqrt{5}$. Square $\sqrt{3}$.

$$\frac{\left(1+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2}+\frac{\left(1-\sqrt{2}\right)\left(\sqrt{5}+\sqrt{3}\right)}{5-3}$$

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

$$\frac{\left(1+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2}+\frac{\left(1-\sqrt{2}\right)\left(\sqrt{5}+\sqrt{3}\right)}{2}$$

Since $\frac{\left(1+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2}$ and $\frac{\left(1-\sqrt{2}\right)\left(\sqrt{5}+\sqrt{3}\right)}{2}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(1+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)+\left(1-\sqrt{2}\right)\left(\sqrt{5}+\sqrt{3}\right)}{2}$$

Do the multiplications in $\left(1+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)+\left(1-\sqrt{2}\right)\left(\sqrt{5}+\sqrt{3}\right)$.

$$\frac{\sqrt{5}-\sqrt{3}+\sqrt{10}-\sqrt{6}+\sqrt{5}+\sqrt{3}-\sqrt{10}-\sqrt{6}}{2}$$

Do the calculations in $\sqrt{5}-\sqrt{3}+\sqrt{10}-\sqrt{6}+\sqrt{5}+\sqrt{3}-\sqrt{10}-\sqrt{6}$.

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

Divide each term of $2\sqrt{5}-2\sqrt{6}$ by $2$ to get $\sqrt{5}-\sqrt{6}$.

$$\sqrt{5}-\sqrt{6}$$