Solve (sqrt{511/49}:sqrt[3]{216/343}+sqrt{140/81})*sqrt[3]{5104/125}

$$( \sqrt { 5 \frac { 11 } { 49 } } : \sqrt[ 3 ] { \frac { 216 } { 343 } } + \sqrt { 1 \frac { 40 } { 81 } } ) \cdot \sqrt[ 3 ] { 5 \frac { 104 } { 125 } }$$

Evaluate

$$7$$

Solution Steps

Multiply 5 and 49 to get 245.

$$\left(\frac{\sqrt{\frac{245+11}{49}}}{\sqrt[3]{\frac{216}{343}}}+\sqrt{\frac{1\times 81+40}{81}}\right)\sqrt[3]{\frac{5\times 125+104}{125}}$$

Add 245 and 11 to get 256.

$$\left(\frac{\sqrt{\frac{256}{49}}}{\sqrt[3]{\frac{216}{343}}}+\sqrt{\frac{1\times 81+40}{81}}\right)\sqrt[3]{\frac{5\times 125+104}{125}}$$

Rewrite the square root of the division \frac{256}{49} as the division of square roots \frac{\sqrt{256}}{\sqrt{49}}. Take the square root of both numerator and denominator.

$$\left(\frac{\frac{16}{7}}{\sqrt[3]{\frac{216}{343}}}+\sqrt{\frac{1\times 81+40}{81}}\right)\sqrt[3]{\frac{5\times 125+104}{125}}$$

Calculate \sqrt[3]{\frac{216}{343}} and get \frac{6}{7}.

$$\left(\frac{\frac{16}{7}}{\frac{6}{7}}+\sqrt{\frac{1\times 81+40}{81}}\right)\sqrt[3]{\frac{5\times 125+104}{125}}$$

Divide \frac{16}{7} by \frac{6}{7} by multiplying \frac{16}{7} by the reciprocal of \frac{6}{7}.

$$\left(\frac{16}{7}\times \frac{7}{6}+\sqrt{\frac{1\times 81+40}{81}}\right)\sqrt[3]{\frac{5\times 125+104}{125}}$$

Multiply \frac{16}{7} and \frac{7}{6} to get \frac{8}{3}.

$$\left(\frac{8}{3}+\sqrt{\frac{1\times 81+40}{81}}\right)\sqrt[3]{\frac{5\times 125+104}{125}}$$

Multiply 1 and 81 to get 81.

$$\left(\frac{8}{3}+\sqrt{\frac{81+40}{81}}\right)\sqrt[3]{\frac{5\times 125+104}{125}}$$

Add 81 and 40 to get 121.

$$\left(\frac{8}{3}+\sqrt{\frac{121}{81}}\right)\sqrt[3]{\frac{5\times 125+104}{125}}$$

Rewrite the square root of the division \frac{121}{81} as the division of square roots \frac{\sqrt{121}}{\sqrt{81}}. Take the square root of both numerator and denominator.

$$\left(\frac{8}{3}+\frac{11}{9}\right)\sqrt[3]{\frac{5\times 125+104}{125}}$$

Add \frac{8}{3} and \frac{11}{9} to get \frac{35}{9}.

$$\frac{35}{9}\sqrt[3]{\frac{5\times 125+104}{125}}$$

Multiply 5 and 125 to get 625.

$$\frac{35}{9}\sqrt[3]{\frac{625+104}{125}}$$

Add 625 and 104 to get 729.

$$\frac{35}{9}\sqrt[3]{\frac{729}{125}}$$

Calculate \sqrt[3]{\frac{729}{125}} and get \frac{9}{5}.

$$\frac{35}{9}\times \frac{9}{5}$$

Multiply \frac{35}{9} and \frac{9}{5} to get 7.

$$7$$

Factor

$$7$$