Solve 1/(4sqrt(2)) * (1 + sqrt(3)) - 4/(4sqrt(2)) * (3 + 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

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

Evaluate

$\frac{-3\sqrt{6}-11\sqrt{2}}{8}\approx -2.863102302$

Short Solution Steps

Rationalize the denominator of $\frac{1}{4\sqrt{2}}$ by multiplying numerator and denominator by $\sqrt{2}$.

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

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

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

Multiply $4$ and $2$ to get $8$.

$$\frac{\sqrt{2}}{8}\left(1+\sqrt{3}\right)-\frac{4}{4\sqrt{2}}\left(3+\sqrt{3}\right)$$

Express $\frac{\sqrt{2}}{8}\left(1+\sqrt{3}\right)$ as a single fraction.

$$\frac{\sqrt{2}\left(1+\sqrt{3}\right)}{8}-\frac{4}{4\sqrt{2}}\left(3+\sqrt{3}\right)$$

Rationalize the denominator of $\frac{4}{4\sqrt{2}}$ by multiplying numerator and denominator by $\sqrt{2}$.

$$\frac{\sqrt{2}\left(1+\sqrt{3}\right)}{8}-\frac{4\sqrt{2}}{4\left(\sqrt{2}\right)^{2}}\left(3+\sqrt{3}\right)$$

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

$$\frac{\sqrt{2}\left(1+\sqrt{3}\right)}{8}-\frac{4\sqrt{2}}{4\times 2}\left(3+\sqrt{3}\right)$$

Cancel out $4$ in both numerator and denominator.

$$\frac{\sqrt{2}\left(1+\sqrt{3}\right)}{8}-\frac{\sqrt{2}}{2}\left(3+\sqrt{3}\right)$$

Express $\frac{\sqrt{2}}{2}\left(3+\sqrt{3}\right)$ as a single fraction.

$$\frac{\sqrt{2}\left(1+\sqrt{3}\right)}{8}-\frac{\sqrt{2}\left(3+\sqrt{3}\right)}{2}$$

Use the distributive property to multiply $\sqrt{2}$ by $3+\sqrt{3}$.

$$\frac{\sqrt{2}\left(1+\sqrt{3}\right)}{8}-\frac{3\sqrt{2}+\sqrt{2}\sqrt{3}}{2}$$

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

$$\frac{\sqrt{2}\left(1+\sqrt{3}\right)}{8}-\frac{3\sqrt{2}+\sqrt{6}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $8$ and $2$ is $8$. Multiply $\frac{3\sqrt{2}+\sqrt{6}}{2}$ times $\frac{4}{4}$.

$$\frac{\sqrt{2}\left(1+\sqrt{3}\right)}{8}-\frac{4\left(3\sqrt{2}+\sqrt{6}\right)}{8}$$

Since $\frac{\sqrt{2}\left(1+\sqrt{3}\right)}{8}$ and $\frac{4\left(3\sqrt{2}+\sqrt{6}\right)}{8}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{2}\left(1+\sqrt{3}\right)-4\left(3\sqrt{2}+\sqrt{6}\right)}{8}$$

Do the multiplications in $\sqrt{2}\left(1+\sqrt{3}\right)-4\left(3\sqrt{2}+\sqrt{6}\right)$.

$$\frac{\sqrt{2}+\sqrt{6}-12\sqrt{2}-4\sqrt{6}}{8}$$

Do the calculations in $\sqrt{2}+\sqrt{6}-12\sqrt{2}-4\sqrt{6}$.

$$\frac{-11\sqrt{2}-3\sqrt{6}}{8}$$

Factor

$\frac{-3 \sqrt{6} - 11 \sqrt{2}}{8} = -2.8631023018066974$