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

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

Evaluate

$$\frac{6\sqrt{6}-\sqrt{3}}{10}\approx 1.296488765$$

Solution Steps

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

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

Calculate the square root of 36 and get 6.

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

Calculate the square root of 9 and get 3.

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

Subtract 3 from 6 to get 3.

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

Multiply 3 and 3 to get 9.

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

Calculate the square root of 9 and get 3.

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

Calculate the square root of 9 and get 3.

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

Multiply 96 and 3 to get 288.

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

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}.

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

Calculate the square root of 4 and get 2.

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

Calculate the square root of 25 and get 5.

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

Multiply 2 and 5 to get 10.

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

Calculate the square root of 9 and get 3.

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

Multiply 5 and 3 to get 15.

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

Add 10 and 15 to get 25.

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

Calculate the square root of 25 and get 5.

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

Calculate the square root of 16 and get 4.

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

Multiply 5 and 4 to get 20.

$$\frac{\sqrt{3}\left(12\sqrt{2}-2\right)}{20}$$

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

$$\frac{12\sqrt{3}\sqrt{2}-2\sqrt{3}}{20}$$

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

$$\frac{12\sqrt{6}-2\sqrt{3}}{20}$$