Solve (5sqrt(2) - sqrt(7))/(8sqrt(3) + 4sqrt(5)) | 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

$$\frac{5\sqrt{2}-\sqrt{7}}{8\sqrt{3}+4\sqrt{5}}$$

Evaluate

$\frac{\sqrt{35}+10\sqrt{6}-2\sqrt{21}-5\sqrt{10}}{28}\approx 0.194087054$

Solution Steps

Rationalize the denominator of $\frac{5\sqrt{2}-\sqrt{7}}{8\sqrt{3}+4\sqrt{5}}$ by multiplying numerator and denominator by $8\sqrt{3}-4\sqrt{5}$.

$$\frac{\left(5\sqrt{2}-\sqrt{7}\right)\left(8\sqrt{3}-4\sqrt{5}\right)}{\left(8\sqrt{3}+4\sqrt{5}\right)\left(8\sqrt{3}-4\sqrt{5}\right)}$$

Consider $\left(8\sqrt{3}+4\sqrt{5}\right)\left(8\sqrt{3}-4\sqrt{5}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{\left(5\sqrt{2}-\sqrt{7}\right)\left(8\sqrt{3}-4\sqrt{5}\right)}{\left(8\sqrt{3}\right)^{2}-\left(4\sqrt{5}\right)^{2}}$$

Expand $\left(8\sqrt{3}\right)^{2}$.

$$\frac{\left(5\sqrt{2}-\sqrt{7}\right)\left(8\sqrt{3}-4\sqrt{5}\right)}{8^{2}\left(\sqrt{3}\right)^{2}-\left(4\sqrt{5}\right)^{2}}$$

Calculate $8$ to the power of $2$ and get $64$.

$$\frac{\left(5\sqrt{2}-\sqrt{7}\right)\left(8\sqrt{3}-4\sqrt{5}\right)}{64\left(\sqrt{3}\right)^{2}-\left(4\sqrt{5}\right)^{2}}$$

The square of $\sqrt{3}$ is $3$.

$$\frac{\left(5\sqrt{2}-\sqrt{7}\right)\left(8\sqrt{3}-4\sqrt{5}\right)}{64\times 3-\left(4\sqrt{5}\right)^{2}}$$

Multiply $64$ and $3$ to get $192$.

$$\frac{\left(5\sqrt{2}-\sqrt{7}\right)\left(8\sqrt{3}-4\sqrt{5}\right)}{192-\left(4\sqrt{5}\right)^{2}}$$

Expand $\left(4\sqrt{5}\right)^{2}$.

$$\frac{\left(5\sqrt{2}-\sqrt{7}\right)\left(8\sqrt{3}-4\sqrt{5}\right)}{192-4^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate $4$ to the power of $2$ and get $16$.

$$\frac{\left(5\sqrt{2}-\sqrt{7}\right)\left(8\sqrt{3}-4\sqrt{5}\right)}{192-16\left(\sqrt{5}\right)^{2}}$$

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

$$\frac{\left(5\sqrt{2}-\sqrt{7}\right)\left(8\sqrt{3}-4\sqrt{5}\right)}{192-16\times 5}$$

Multiply $16$ and $5$ to get $80$.

$$\frac{\left(5\sqrt{2}-\sqrt{7}\right)\left(8\sqrt{3}-4\sqrt{5}\right)}{192-80}$$

Subtract $80$ from $192$ to get $112$.

$$\frac{\left(5\sqrt{2}-\sqrt{7}\right)\left(8\sqrt{3}-4\sqrt{5}\right)}{112}$$

Apply the distributive property by multiplying each term of $5\sqrt{2}-\sqrt{7}$ by each term of $8\sqrt{3}-4\sqrt{5}$.

$$\frac{40\sqrt{3}\sqrt{2}-20\sqrt{2}\sqrt{5}-8\sqrt{3}\sqrt{7}+4\sqrt{7}\sqrt{5}}{112}$$

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

$$\frac{40\sqrt{6}-20\sqrt{2}\sqrt{5}-8\sqrt{3}\sqrt{7}+4\sqrt{7}\sqrt{5}}{112}$$

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

$$\frac{40\sqrt{6}-20\sqrt{10}-8\sqrt{3}\sqrt{7}+4\sqrt{7}\sqrt{5}}{112}$$

To multiply $\sqrt{3}$ and $\sqrt{7}$, multiply the numbers under the square root.

$$\frac{40\sqrt{6}-20\sqrt{10}-8\sqrt{21}+4\sqrt{7}\sqrt{5}}{112}$$

To multiply $\sqrt{7}$ and $\sqrt{5}$, multiply the numbers under the square root.

$$\frac{40\sqrt{6}-20\sqrt{10}-8\sqrt{21}+4\sqrt{35}}{112}$$