Solve frac{sqrt{--5^2+sqrt[3]{-8^2}}}{-sqrt{-5*-8}}

$$\frac{ \sqrt{ -- { 5 }^{ 2 } + \sqrt[ 3 ]{ { -8 }^{ 2 } } } }{ - \sqrt{ -5 \times -8 } }$$

Evaluate

$$-\frac{\sqrt{290}}{20}\approx -0.851469318$$

Solution Steps

Calculate 5 to the power of 2 and get 25.

$$\frac{\sqrt{-\left(-25\right)+\sqrt[3]{\left(-8\right)^{2}}}}{-\sqrt{-5\left(-8\right)}}$$

The opposite of -25 is 25.

$$\frac{\sqrt{25+\sqrt[3]{\left(-8\right)^{2}}}}{-\sqrt{-5\left(-8\right)}}$$

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

$$\frac{\sqrt{25+\sqrt[3]{64}}}{-\sqrt{-5\left(-8\right)}}$$

Calculate \sqrt[3]{64} and get 4.

$$\frac{\sqrt{25+4}}{-\sqrt{-5\left(-8\right)}}$$

Add 25 and 4 to get 29.

$$\frac{\sqrt{29}}{-\sqrt{-5\left(-8\right)}}$$

Multiply -5 and -8 to get 40.

$$\frac{\sqrt{29}}{-\sqrt{40}}$$

Factor 40=2^{2}\times 10. Rewrite the square root of the product \sqrt{2^{2}\times 10} as the product of square roots \sqrt{2^{2}}\sqrt{10}. Take the square root of 2^{2}.

$$\frac{\sqrt{29}}{-2\sqrt{10}}$$

Rationalize the denominator of \frac{\sqrt{29}}{-2\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{\sqrt{29}\sqrt{10}}{-2\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{\sqrt{29}\sqrt{10}}{-2\times 10}$$

To multiply \sqrt{29} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{\sqrt{290}}{-2\times 10}$$

Multiply -2 and 10 to get -20.

$$\frac{\sqrt{290}}{-20}$$