Solve frac{sqrt{3}(sqrt{36}-sqrt{9})}{sqrt{2sqrt{25}+5sqrt{9}}}divfrac{sqrt{sqrt{16}}}{sqrt{96sqrt{9}}-sqrt{4}}

$$\frac { \sqrt { 3 } ( \sqrt { 36 } - \sqrt { 9 } ) } { \sqrt { 2 \sqrt { 25 } + 5 \sqrt { 9 } } } \div \frac { \sqrt { \sqrt { 16 } } } { \sqrt { 96 \sqrt { 9 } } - \sqrt { 4 } }$$

Evaluate

$$\frac{18\sqrt{6}-3\sqrt{3}}{5}\approx 7.778932589$$

Solution Steps

Divide \frac{\sqrt{3}\left(\sqrt{36}-\sqrt{9}\right)}{\sqrt{2\sqrt{25}+5\sqrt{9}}} by \frac{\sqrt{\sqrt{16}}}{\sqrt{96\sqrt{9}}-\sqrt{4}} by multiplying \frac{\sqrt{3}\left(\sqrt{36}-\sqrt{9}\right)}{\sqrt{2\sqrt{25}+5\sqrt{9}}} by the reciprocal of \frac{\sqrt{\sqrt{16}}}{\sqrt{96\sqrt{9}}-\sqrt{4}}.

$$\frac{\sqrt{3}\left(\sqrt{36}-\sqrt{9}\right)\left(\sqrt{96\sqrt{9}}-\sqrt{4}\right)}{\sqrt{2\sqrt{25}+5\sqrt{9}}\sqrt{\sqrt{16}}}$$

Calculate the square root of 36 and get 6.

$$\frac{\sqrt{3}\left(6-\sqrt{9}\right)\left(\sqrt{96\sqrt{9}}-\sqrt{4}\right)}{\sqrt{2\sqrt{25}+5\sqrt{9}}\sqrt{\sqrt{16}}}$$

Calculate the square root of 9 and get 3.

$$\frac{\sqrt{3}\left(6-3\right)\left(\sqrt{96\sqrt{9}}-\sqrt{4}\right)}{\sqrt{2\sqrt{25}+5\sqrt{9}}\sqrt{\sqrt{16}}}$$

Subtract 3 from 6 to get 3.

$$\frac{\sqrt{3}\times 3\left(\sqrt{96\sqrt{9}}-\sqrt{4}\right)}{\sqrt{2\sqrt{25}+5\sqrt{9}}\sqrt{\sqrt{16}}}$$

Calculate the square root of 9 and get 3.

$$\frac{\sqrt{3}\times 3\left(\sqrt{96\times 3}-\sqrt{4}\right)}{\sqrt{2\sqrt{25}+5\sqrt{9}}\sqrt{\sqrt{16}}}$$

Multiply 96 and 3 to get 288.

$$\frac{\sqrt{3}\times 3\left(\sqrt{288}-\sqrt{4}\right)}{\sqrt{2\sqrt{25}+5\sqrt{9}}\sqrt{\sqrt{16}}}$$

Factor 288=12^{2}\times 2. Rewrite the square root of the product \sqrt{12^{2}\times 2} as the product of square roots \sqrt{12^{2}}\sqrt{2}. Take the square root of 12^{2}.

$$\frac{\sqrt{3}\times 3\left(12\sqrt{2}-\sqrt{4}\right)}{\sqrt{2\sqrt{25}+5\sqrt{9}}\sqrt{\sqrt{16}}}$$

Calculate the square root of 4 and get 2.

$$\frac{\sqrt{3}\times 3\left(12\sqrt{2}-2\right)}{\sqrt{2\sqrt{25}+5\sqrt{9}}\sqrt{\sqrt{16}}}$$

Calculate the square root of 25 and get 5.

$$\frac{\sqrt{3}\times 3\left(12\sqrt{2}-2\right)}{\sqrt{2\times 5+5\sqrt{9}}\sqrt{\sqrt{16}}}$$

Multiply 2 and 5 to get 10.

$$\frac{\sqrt{3}\times 3\left(12\sqrt{2}-2\right)}{\sqrt{10+5\sqrt{9}}\sqrt{\sqrt{16}}}$$

Calculate the square root of 9 and get 3.

$$\frac{\sqrt{3}\times 3\left(12\sqrt{2}-2\right)}{\sqrt{10+5\times 3}\sqrt{\sqrt{16}}}$$

Multiply 5 and 3 to get 15.

$$\frac{\sqrt{3}\times 3\left(12\sqrt{2}-2\right)}{\sqrt{10+15}\sqrt{\sqrt{16}}}$$

Add 10 and 15 to get 25.

$$\frac{\sqrt{3}\times 3\left(12\sqrt{2}-2\right)}{\sqrt{25}\sqrt{\sqrt{16}}}$$

Calculate the square root of 25 and get 5.

$$\frac{\sqrt{3}\times 3\left(12\sqrt{2}-2\right)}{5\sqrt{\sqrt{16}}}$$

Calculate the square root of 16 and get 4.

$$\frac{\sqrt{3}\times 3\left(12\sqrt{2}-2\right)}{5\sqrt{4}}$$

Calculate the square root of 4 and get 2.

$$\frac{\sqrt{3}\times 3\left(12\sqrt{2}-2\right)}{5\times 2}$$

Multiply 5 and 2 to get 10.

$$\frac{\sqrt{3}\times 3\left(12\sqrt{2}-2\right)}{10}$$

Use the distributive property to multiply \sqrt{3}\times 3 by 12\sqrt{2}-2.

$$\frac{36\sqrt{3}\sqrt{2}-2\sqrt{3}\times 3}{10}$$

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

$$\frac{36\sqrt{6}-2\sqrt{3}\times 3}{10}$$

Multiply -2 and 3 to get -6.

$$\frac{36\sqrt{6}-6\sqrt{3}}{10}$$