Solve sqrt{sqrt[3]{125}+frac{-sqrt{64}}{sqrt{144}}}+sqrt[3]{1000}

$$\sqrt{ \sqrt[ 3 ]{ 125 } + \frac{ - \sqrt{ 64 } }{ \sqrt{ 144 } } } + \sqrt[ 3 ]{ 1000 }$$

Evaluate

$$\frac{\sqrt{39}}{3}+10\approx 12.081665999$$

Short Solution Steps

Calculate \sqrt[3]{125} and get 5.

$$\sqrt{5+\frac{-\sqrt{64}}{\sqrt{144}}}+\sqrt[3]{1000}$$

Calculate the square root of 64 and get 8.

$$\sqrt{5+\frac{-8}{\sqrt{144}}}+\sqrt[3]{1000}$$

Calculate the square root of 144 and get 12.

$$\sqrt{5+\frac{-8}{12}}+\sqrt[3]{1000}$$

Reduce the fraction \frac{-8}{12} to lowest terms by extracting and canceling out 4.

$$\sqrt{5-\frac{2}{3}}+\sqrt[3]{1000}$$

Subtract \frac{2}{3} from 5 to get \frac{13}{3}.

$$\sqrt{\frac{13}{3}}+\sqrt[3]{1000}$$

Rewrite the square root of the division \sqrt{\frac{13}{3}} as the division of square roots \frac{\sqrt{13}}{\sqrt{3}}.

$$\frac{\sqrt{13}}{\sqrt{3}}+\sqrt[3]{1000}$$

Rationalize the denominator of \frac{\sqrt{13}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\sqrt{13}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\sqrt[3]{1000}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{13}\sqrt{3}}{3}+\sqrt[3]{1000}$$

To multiply \sqrt{13} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\sqrt{39}}{3}+\sqrt[3]{1000}$$

Calculate \sqrt[3]{1000} and get 10.

$$\frac{\sqrt{39}}{3}+10$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 10 times \frac{3}{3}.

$$\frac{\sqrt{39}}{3}+\frac{10\times 3}{3}$$

Since \frac{\sqrt{39}}{3} and \frac{10\times 3}{3} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{39}+10\times 3}{3}$$

Do the multiplications in \sqrt{39}+10\times 3.

$$\frac{\sqrt{39}+30}{3}$$