Solve (2sqrt(2) + 3sqrt(5))/(sqrt(2) - sqrt(6)) | 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

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

Evaluate

$-\frac{3\sqrt{10}}{4}-\frac{3\sqrt{30}}{4}-\sqrt{3}-1\approx -9.211678234$

Solution Steps

Rationalize the denominator of $\frac{2\sqrt{2}+3\sqrt{5}}{\sqrt{2}-\sqrt{6}}$ by multiplying numerator and denominator by $\sqrt{2}+\sqrt{6}$.

$$\frac{\left(2\sqrt{2}+3\sqrt{5}\right)\left(\sqrt{2}+\sqrt{6}\right)}{\left(\sqrt{2}-\sqrt{6}\right)\left(\sqrt{2}+\sqrt{6}\right)}$$

Consider $\left(\sqrt{2}-\sqrt{6}\right)\left(\sqrt{2}+\sqrt{6}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{\left(2\sqrt{2}+3\sqrt{5}\right)\left(\sqrt{2}+\sqrt{6}\right)}{\left(\sqrt{2}\right)^{2}-\left(\sqrt{6}\right)^{2}}$$

Square $\sqrt{2}$. Square $\sqrt{6}$.

$$\frac{\left(2\sqrt{2}+3\sqrt{5}\right)\left(\sqrt{2}+\sqrt{6}\right)}{2-6}$$

Subtract $6$ from $2$ to get $-4$.

$$\frac{\left(2\sqrt{2}+3\sqrt{5}\right)\left(\sqrt{2}+\sqrt{6}\right)}{-4}$$

Apply the distributive property by multiplying each term of $2\sqrt{2}+3\sqrt{5}$ by each term of $\sqrt{2}+\sqrt{6}$.

$$\frac{2\left(\sqrt{2}\right)^{2}+2\sqrt{2}\sqrt{6}+3\sqrt{5}\sqrt{2}+3\sqrt{5}\sqrt{6}}{-4}$$

The square of $\sqrt{2}$ is $2$.

$$\frac{2\times 2+2\sqrt{2}\sqrt{6}+3\sqrt{5}\sqrt{2}+3\sqrt{5}\sqrt{6}}{-4}$$

Multiply $2$ and $2$ to get $4$.

$$\frac{4+2\sqrt{2}\sqrt{6}+3\sqrt{5}\sqrt{2}+3\sqrt{5}\sqrt{6}}{-4}$$

Factor $6=2\times 3$. Rewrite the square root of the product $\sqrt{2\times 3}$ as the product of square roots $\sqrt{2}\sqrt{3}$.

$$\frac{4+2\sqrt{2}\sqrt{2}\sqrt{3}+3\sqrt{5}\sqrt{2}+3\sqrt{5}\sqrt{6}}{-4}$$

Multiply $\sqrt{2}$ and $\sqrt{2}$ to get $2$.

$$\frac{4+2\times 2\sqrt{3}+3\sqrt{5}\sqrt{2}+3\sqrt{5}\sqrt{6}}{-4}$$

Multiply $2$ and $2$ to get $4$.

$$\frac{4+4\sqrt{3}+3\sqrt{5}\sqrt{2}+3\sqrt{5}\sqrt{6}}{-4}$$

To multiply $\sqrt{5}$ and $\sqrt{2}$, multiply the numbers under the square root.

$$\frac{4+4\sqrt{3}+3\sqrt{10}+3\sqrt{5}\sqrt{6}}{-4}$$

To multiply $\sqrt{5}$ and $\sqrt{6}$, multiply the numbers under the square root.

$$\frac{4+4\sqrt{3}+3\sqrt{10}+3\sqrt{30}}{-4}$$

Multiply both numerator and denominator by -1.

$$\frac{-4-4\sqrt{3}-3\sqrt{10}-3\sqrt{30}}{4}$$