Solve frac{8sqrt{5}}{6sqrt{2}-7sqrt{10}}

$$\frac{ 8 \sqrt{ 5 } }{ 6 \sqrt{ 2 } -7 \sqrt{ 10 } }$$

Evaluate

$$\frac{-24\sqrt{10}-140\sqrt{2}}{209}\approx -1.310452453$$

Solution Steps

Rationalize the denominator of \frac{8\sqrt{5}}{6\sqrt{2}-7\sqrt{10}} by multiplying numerator and denominator by 6\sqrt{2}+7\sqrt{10}.

$$\frac{8\sqrt{5}\left(6\sqrt{2}+7\sqrt{10}\right)}{\left(6\sqrt{2}-7\sqrt{10}\right)\left(6\sqrt{2}+7\sqrt{10}\right)}$$

Consider \left(6\sqrt{2}-7\sqrt{10}\right)\left(6\sqrt{2}+7\sqrt{10}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{8\sqrt{5}\left(6\sqrt{2}+7\sqrt{10}\right)}{\left(6\sqrt{2}\right)^{2}-\left(-7\sqrt{10}\right)^{2}}$$

Expand \left(6\sqrt{2}\right)^{2}.

$$\frac{8\sqrt{5}\left(6\sqrt{2}+7\sqrt{10}\right)}{6^{2}\left(\sqrt{2}\right)^{2}-\left(-7\sqrt{10}\right)^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{8\sqrt{5}\left(6\sqrt{2}+7\sqrt{10}\right)}{36\left(\sqrt{2}\right)^{2}-\left(-7\sqrt{10}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{8\sqrt{5}\left(6\sqrt{2}+7\sqrt{10}\right)}{36\times 2-\left(-7\sqrt{10}\right)^{2}}$$

Multiply 36 and 2 to get 72.

$$\frac{8\sqrt{5}\left(6\sqrt{2}+7\sqrt{10}\right)}{72-\left(-7\sqrt{10}\right)^{2}}$$

Expand \left(-7\sqrt{10}\right)^{2}.

$$\frac{8\sqrt{5}\left(6\sqrt{2}+7\sqrt{10}\right)}{72-\left(-7\right)^{2}\left(\sqrt{10}\right)^{2}}$$

Calculate -7 to the power of 2 and get 49.

$$\frac{8\sqrt{5}\left(6\sqrt{2}+7\sqrt{10}\right)}{72-49\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{8\sqrt{5}\left(6\sqrt{2}+7\sqrt{10}\right)}{72-49\times 10}$$

Multiply 49 and 10 to get 490.

$$\frac{8\sqrt{5}\left(6\sqrt{2}+7\sqrt{10}\right)}{72-490}$$

Subtract 490 from 72 to get -418.

$$\frac{8\sqrt{5}\left(6\sqrt{2}+7\sqrt{10}\right)}{-418}$$

Divide 8\sqrt{5}\left(6\sqrt{2}+7\sqrt{10}\right) by -418 to get -\frac{4}{209}\sqrt{5}\left(6\sqrt{2}+7\sqrt{10}\right).

$$-\frac{4}{209}\sqrt{5}\left(6\sqrt{2}+7\sqrt{10}\right)$$

Use the distributive property to multiply -\frac{4}{209}\sqrt{5} by 6\sqrt{2}+7\sqrt{10}.

$$-\frac{4}{209}\sqrt{5}\times 6\sqrt{2}-\frac{4}{209}\sqrt{5}\times 7\sqrt{10}$$

Express -\frac{4}{209}\times 6 as a single fraction.

$$\frac{-4\times 6}{209}\sqrt{5}\sqrt{2}-\frac{4}{209}\sqrt{5}\times 7\sqrt{10}$$

Multiply -4 and 6 to get -24.

$$\frac{-24}{209}\sqrt{5}\sqrt{2}-\frac{4}{209}\sqrt{5}\times 7\sqrt{10}$$

Fraction \frac{-24}{209} can be rewritten as -\frac{24}{209} by extracting the negative sign.

$$-\frac{24}{209}\sqrt{5}\sqrt{2}-\frac{4}{209}\sqrt{5}\times 7\sqrt{10}$$

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

$$-\frac{24}{209}\sqrt{10}-\frac{4}{209}\sqrt{5}\times 7\sqrt{10}$$

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

$$-\frac{24}{209}\sqrt{10}-\frac{4}{209}\sqrt{5}\times 7\sqrt{5}\sqrt{2}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$-\frac{24}{209}\sqrt{10}-\frac{4}{209}\times 5\times 7\sqrt{2}$$

Express -\frac{4}{209}\times 5 as a single fraction.

$$-\frac{24}{209}\sqrt{10}+\frac{-4\times 5}{209}\times 7\sqrt{2}$$

Multiply -4 and 5 to get -20.

$$-\frac{24}{209}\sqrt{10}+\frac{-20}{209}\times 7\sqrt{2}$$

Fraction \frac{-20}{209} can be rewritten as -\frac{20}{209} by extracting the negative sign.

$$-\frac{24}{209}\sqrt{10}-\frac{20}{209}\times 7\sqrt{2}$$

Express -\frac{20}{209}\times 7 as a single fraction.

$$-\frac{24}{209}\sqrt{10}+\frac{-20\times 7}{209}\sqrt{2}$$

Multiply -20 and 7 to get -140.

$$-\frac{24}{209}\sqrt{10}+\frac{-140}{209}\sqrt{2}$$

Fraction \frac{-140}{209} can be rewritten as -\frac{140}{209} by extracting the negative sign.

$$-\frac{24}{209}\sqrt{10}-\frac{140}{209}\sqrt{2}$$