Solve 10*10-10*20/sqrt{10*20-10^2sqrt{10*68-20^2}}

$$\frac{ 10 \times 10-10 \times 20 }{ \sqrt{ 10 \times 20- { 10 }^{ 2 } } \sqrt{ 10 \times 68- { 20 }^{ 2 } } }$$

Evaluate

$$-\frac{\sqrt{70}}{14}\approx -0.597614305$$

Solution Steps

Multiply 10 and 10 to get 100.

$$\frac{100-10\times 20}{\sqrt{10\times 20-10^{2}}\sqrt{10\times 68-20^{2}}}$$

Multiply 10 and 20 to get 200.

$$\frac{100-200}{\sqrt{10\times 20-10^{2}}\sqrt{10\times 68-20^{2}}}$$

Subtract 200 from 100 to get -100.

$$\frac{-100}{\sqrt{10\times 20-10^{2}}\sqrt{10\times 68-20^{2}}}$$

Multiply 10 and 20 to get 200.

$$\frac{-100}{\sqrt{200-10^{2}}\sqrt{10\times 68-20^{2}}}$$

Calculate 10 to the power of 2 and get 100.

$$\frac{-100}{\sqrt{200-100}\sqrt{10\times 68-20^{2}}}$$

Subtract 100 from 200 to get 100.

$$\frac{-100}{\sqrt{100}\sqrt{10\times 68-20^{2}}}$$

Calculate the square root of 100 and get 10.

$$\frac{-100}{10\sqrt{10\times 68-20^{2}}}$$

Multiply 10 and 68 to get 680.

$$\frac{-100}{10\sqrt{680-20^{2}}}$$

Calculate 20 to the power of 2 and get 400.

$$\frac{-100}{10\sqrt{680-400}}$$

Subtract 400 from 680 to get 280.

$$\frac{-100}{10\sqrt{280}}$$

Factor 280=2^{2}\times 70. Rewrite the square root of the product \sqrt{2^{2}\times 70} as the product of square roots \sqrt{2^{2}}\sqrt{70}. Take the square root of 2^{2}.

$$\frac{-100}{10\times 2\sqrt{70}}$$

Multiply 10 and 2 to get 20.

$$\frac{-100}{20\sqrt{70}}$$

Rationalize the denominator of \frac{-100}{20\sqrt{70}} by multiplying numerator and denominator by \sqrt{70}.

$$\frac{-100\sqrt{70}}{20\left(\sqrt{70}\right)^{2}}$$

The square of \sqrt{70} is 70.

$$\frac{-100\sqrt{70}}{20\times 70}$$

Cancel out 5\times 20 in both numerator and denominator.

$$\frac{-\sqrt{70}}{14}$$