Solve sqrt{16/15:7/8-13/15:13/10+frac{1}{3}*frac{5}{3}}=

$$\sqrt { \frac { 16 } { 15 } : \frac { 7 } { 8 } - \frac { 13 } { 15 } : \frac { 13 } { 10 } + \frac { 1 } { 3 } \times \frac { 5 } { 3 } } =$$

Evaluate

$$\frac{\sqrt{12215}}{105}\approx 1.05258563$$

Solution Steps

Divide \frac{16}{15} by \frac{7}{8} by multiplying \frac{16}{15} by the reciprocal of \frac{7}{8}.

$$\sqrt{\frac{16}{15}\times \frac{8}{7}-\frac{\frac{13}{15}}{\frac{13}{10}}+\frac{1}{3}\times \frac{5}{3}}$$

Multiply \frac{16}{15} times \frac{8}{7} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{16\times 8}{15\times 7}-\frac{\frac{13}{15}}{\frac{13}{10}}+\frac{1}{3}\times \frac{5}{3}}$$

Do the multiplications in the fraction \frac{16\times 8}{15\times 7}.

$$\sqrt{\frac{128}{105}-\frac{\frac{13}{15}}{\frac{13}{10}}+\frac{1}{3}\times \frac{5}{3}}$$

Divide \frac{13}{15} by \frac{13}{10} by multiplying \frac{13}{15} by the reciprocal of \frac{13}{10}.

$$\sqrt{\frac{128}{105}-\frac{13}{15}\times \frac{10}{13}+\frac{1}{3}\times \frac{5}{3}}$$

Multiply \frac{13}{15} times \frac{10}{13} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{128}{105}-\frac{13\times 10}{15\times 13}+\frac{1}{3}\times \frac{5}{3}}$$

Cancel out 13 in both numerator and denominator.

$$\sqrt{\frac{128}{105}-\frac{10}{15}+\frac{1}{3}\times \frac{5}{3}}$$

Reduce the fraction \frac{10}{15} to lowest terms by extracting and canceling out 5.

$$\sqrt{\frac{128}{105}-\frac{2}{3}+\frac{1}{3}\times \frac{5}{3}}$$

Least common multiple of 105 and 3 is 105. Convert \frac{128}{105} and \frac{2}{3} to fractions with denominator 105.

$$\sqrt{\frac{128}{105}-\frac{70}{105}+\frac{1}{3}\times \frac{5}{3}}$$

Since \frac{128}{105} and \frac{70}{105} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{128-70}{105}+\frac{1}{3}\times \frac{5}{3}}$$

Subtract 70 from 128 to get 58.

$$\sqrt{\frac{58}{105}+\frac{1}{3}\times \frac{5}{3}}$$

Multiply \frac{1}{3} times \frac{5}{3} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{58}{105}+\frac{1\times 5}{3\times 3}}$$

Do the multiplications in the fraction \frac{1\times 5}{3\times 3}.

$$\sqrt{\frac{58}{105}+\frac{5}{9}}$$

Least common multiple of 105 and 9 is 315. Convert \frac{58}{105} and \frac{5}{9} to fractions with denominator 315.

$$\sqrt{\frac{174}{315}+\frac{175}{315}}$$

Since \frac{174}{315} and \frac{175}{315} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{174+175}{315}}$$

Add 174 and 175 to get 349.

$$\sqrt{\frac{349}{315}}$$

Rewrite the square root of the division \sqrt{\frac{349}{315}} as the division of square roots \frac{\sqrt{349}}{\sqrt{315}}.

$$\frac{\sqrt{349}}{\sqrt{315}}$$

Factor 315=3^{2}\times 35. Rewrite the square root of the product \sqrt{3^{2}\times 35} as the product of square roots \sqrt{3^{2}}\sqrt{35}. Take the square root of 3^{2}.

$$\frac{\sqrt{349}}{3\sqrt{35}}$$

Rationalize the denominator of \frac{\sqrt{349}}{3\sqrt{35}} by multiplying numerator and denominator by \sqrt{35}.

$$\frac{\sqrt{349}\sqrt{35}}{3\left(\sqrt{35}\right)^{2}}$$

The square of \sqrt{35} is 35.

$$\frac{\sqrt{349}\sqrt{35}}{3\times 35}$$

To multiply \sqrt{349} and \sqrt{35}, multiply the numbers under the square root.

$$\frac{\sqrt{12215}}{3\times 35}$$

Multiply 3 and 35 to get 105.

$$\frac{\sqrt{12215}}{105}$$