Solve {l}{a=frac{sqrt{5}}{sqrt{3}}*sqrt{6/5}*frac{sqrt{15}}{sqrt{10}}}{b=sqrt{3/4}divsqrt{0.6}divsqrt{5/28}}

$$\left. \begin{array} { l } { a = \frac { \sqrt { 5 } } { \sqrt { 3 } } \times \sqrt { \frac { 6 } { 5 } } \times \frac { \sqrt { 15 } } { \sqrt { 10 } } } \\ { b = \sqrt { \frac { 3 } { 4 } } \div \sqrt { 0.6 } \div \sqrt { \frac { 5 } { 28 } } } \end{array} \right.$$

Solve for a, b

$$a=\sqrt{3}\approx 1.732050808$$

$$b=\sqrt{7}\approx 2.645751311$$

Solution Steps

Consider the first equation. Rationalize the denominator of \frac{\sqrt{5}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$a=\frac{\sqrt{5}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\sqrt{\frac{6}{5}}\times \frac{\sqrt{15}}{\sqrt{10}}$$

The square of \sqrt{3} is 3.

$$a=\frac{\sqrt{5}\sqrt{3}}{3}\sqrt{\frac{6}{5}}\times \frac{\sqrt{15}}{\sqrt{10}}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$a=\frac{\sqrt{15}}{3}\sqrt{\frac{6}{5}}\times \frac{\sqrt{15}}{\sqrt{10}}$$

Rewrite the square root of the division \sqrt{\frac{6}{5}} as the division of square roots \frac{\sqrt{6}}{\sqrt{5}}.

$$a=\frac{\sqrt{15}}{3}\times \frac{\sqrt{6}}{\sqrt{5}}\times \frac{\sqrt{15}}{\sqrt{10}}$$

Rationalize the denominator of \frac{\sqrt{6}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$a=\frac{\sqrt{15}}{3}\times \frac{\sqrt{6}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\times \frac{\sqrt{15}}{\sqrt{10}}$$

The square of \sqrt{5} is 5.

$$a=\frac{\sqrt{15}}{3}\times \frac{\sqrt{6}\sqrt{5}}{5}\times \frac{\sqrt{15}}{\sqrt{10}}$$

To multiply \sqrt{6} and \sqrt{5}, multiply the numbers under the square root.

$$a=\frac{\sqrt{15}}{3}\times \frac{\sqrt{30}}{5}\times \frac{\sqrt{15}}{\sqrt{10}}$$

Rationalize the denominator of \frac{\sqrt{15}}{\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$a=\frac{\sqrt{15}}{3}\times \frac{\sqrt{30}}{5}\times \frac{\sqrt{15}\sqrt{10}}{\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$a=\frac{\sqrt{15}}{3}\times \frac{\sqrt{30}}{5}\times \frac{\sqrt{15}\sqrt{10}}{10}$$

To multiply \sqrt{15} and \sqrt{10}, multiply the numbers under the square root.

$$a=\frac{\sqrt{15}}{3}\times \frac{\sqrt{30}}{5}\times \frac{\sqrt{150}}{10}$$

Factor 150=5^{2}\times 6. Rewrite the square root of the product \sqrt{5^{2}\times 6} as the product of square roots \sqrt{5^{2}}\sqrt{6}. Take the square root of 5^{2}.

$$a=\frac{\sqrt{15}}{3}\times \frac{\sqrt{30}}{5}\times \frac{5\sqrt{6}}{10}$$

Divide 5\sqrt{6} by 10 to get \frac{1}{2}\sqrt{6}.

$$a=\frac{\sqrt{15}}{3}\times \frac{\sqrt{30}}{5}\times \frac{1}{2}\sqrt{6}$$

Multiply \frac{\sqrt{15}}{3} times \frac{\sqrt{30}}{5} by multiplying numerator times numerator and denominator times denominator.

$$a=\frac{\sqrt{15}\sqrt{30}}{3\times 5}\times \frac{1}{2}\sqrt{6}$$

Multiply \frac{\sqrt{15}\sqrt{30}}{3\times 5} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$a=\frac{\sqrt{15}\sqrt{30}}{3\times 5\times 2}\sqrt{6}$$

Express \frac{\sqrt{15}\sqrt{30}}{3\times 5\times 2}\sqrt{6} as a single fraction.

$$a=\frac{\sqrt{15}\sqrt{30}\sqrt{6}}{3\times 5\times 2}$$

Factor 30=15\times 2. Rewrite the square root of the product \sqrt{15\times 2} as the product of square roots \sqrt{15}\sqrt{2}.

$$a=\frac{\sqrt{15}\sqrt{15}\sqrt{2}\sqrt{6}}{3\times 5\times 2}$$

Multiply \sqrt{15} and \sqrt{15} to get 15.

$$a=\frac{15\sqrt{2}\sqrt{6}}{3\times 5\times 2}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$a=\frac{15\sqrt{2}\sqrt{2}\sqrt{3}}{3\times 5\times 2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$a=\frac{15\times 2\sqrt{3}}{3\times 5\times 2}$$

Multiply 15 and 2 to get 30.

$$a=\frac{30\sqrt{3}}{3\times 5\times 2}$$

Multiply 3 and 5 to get 15.

$$a=\frac{30\sqrt{3}}{15\times 2}$$

Multiply 15 and 2 to get 30.

$$a=\frac{30\sqrt{3}}{30}$$

Cancel out 30 and 30.

$$a=\sqrt{3}$$

Consider the second equation. Rewrite the square root of the division \sqrt{\frac{3}{4}} as the division of square roots \frac{\sqrt{3}}{\sqrt{4}}.

$$b=\frac{\frac{\frac{\sqrt{3}}{\sqrt{4}}}{\sqrt{0.6}}}{\sqrt{\frac{5}{28}}}$$

Calculate the square root of 4 and get 2.

$$b=\frac{\frac{\frac{\sqrt{3}}{2}}{\sqrt{0.6}}}{\sqrt{\frac{5}{28}}}$$

Rationalize the denominator of \frac{\frac{\sqrt{3}}{2}}{\sqrt{0.6}} by multiplying numerator and denominator by \sqrt{0.6}.

$$b=\frac{\frac{\frac{\sqrt{3}}{2}\sqrt{0.6}}{\left(\sqrt{0.6}\right)^{2}}}{\sqrt{\frac{5}{28}}}$$

The square of \sqrt{0.6} is 0.6.

$$b=\frac{\frac{\frac{\sqrt{3}}{2}\sqrt{0.6}}{0.6}}{\sqrt{\frac{5}{28}}}$$

Rewrite the square root of the division \sqrt{\frac{5}{28}} as the division of square roots \frac{\sqrt{5}}{\sqrt{28}}.

$$b=\frac{\frac{\frac{\sqrt{3}}{2}\sqrt{0.6}}{0.6}}{\frac{\sqrt{5}}{\sqrt{28}}}$$

Factor 28=2^{2}\times 7. Rewrite the square root of the product \sqrt{2^{2}\times 7} as the product of square roots \sqrt{2^{2}}\sqrt{7}. Take the square root of 2^{2}.

$$b=\frac{\frac{\frac{\sqrt{3}}{2}\sqrt{0.6}}{0.6}}{\frac{\sqrt{5}}{2\sqrt{7}}}$$

Rationalize the denominator of \frac{\sqrt{5}}{2\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$b=\frac{\frac{\frac{\sqrt{3}}{2}\sqrt{0.6}}{0.6}}{\frac{\sqrt{5}\sqrt{7}}{2\left(\sqrt{7}\right)^{2}}}$$

The square of \sqrt{7} is 7.

$$b=\frac{\frac{\frac{\sqrt{3}}{2}\sqrt{0.6}}{0.6}}{\frac{\sqrt{5}\sqrt{7}}{2\times 7}}$$

To multiply \sqrt{5} and \sqrt{7}, multiply the numbers under the square root.

$$b=\frac{\frac{\frac{\sqrt{3}}{2}\sqrt{0.6}}{0.6}}{\frac{\sqrt{35}}{2\times 7}}$$

Multiply 2 and 7 to get 14.

$$b=\frac{\frac{\frac{\sqrt{3}}{2}\sqrt{0.6}}{0.6}}{\frac{\sqrt{35}}{14}}$$

Divide \frac{\frac{\sqrt{3}}{2}\sqrt{0.6}}{0.6} by \frac{\sqrt{35}}{14} by multiplying \frac{\frac{\sqrt{3}}{2}\sqrt{0.6}}{0.6} by the reciprocal of \frac{\sqrt{35}}{14}.

$$b=\frac{\frac{\sqrt{3}}{2}\sqrt{0.6}\times 14}{0.6\sqrt{35}}$$

Rationalize the denominator of \frac{\frac{\sqrt{3}}{2}\sqrt{0.6}\times 14}{0.6\sqrt{35}} by multiplying numerator and denominator by \sqrt{35}.

$$b=\frac{\frac{\sqrt{3}}{2}\sqrt{0.6}\times 14\sqrt{35}}{0.6\left(\sqrt{35}\right)^{2}}$$

The square of \sqrt{35} is 35.

$$b=\frac{\frac{\sqrt{3}}{2}\sqrt{0.6}\times 14\sqrt{35}}{0.6\times 35}$$

Cancel out 2, the greatest common factor in 14 and 2.

$$b=\frac{7\sqrt{3}\sqrt{0.6}\sqrt{35}}{0.6\times 35}$$

To multiply \sqrt{3} and \sqrt{0.6}, multiply the numbers under the square root.

$$b=\frac{7\sqrt{1.8}\sqrt{35}}{0.6\times 35}$$

To multiply \sqrt{1.8} and \sqrt{35}, multiply the numbers under the square root.

$$b=\frac{7\sqrt{63}}{0.6\times 35}$$

Multiply 0.6 and 35 to get 21.

$$b=\frac{7\sqrt{63}}{21}$$

Factor 63=3^{2}\times 7. Rewrite the square root of the product \sqrt{3^{2}\times 7} as the product of square roots \sqrt{3^{2}}\sqrt{7}. Take the square root of 3^{2}.

$$b=\frac{7\times 3\sqrt{7}}{21}$$

Multiply 7 and 3 to get 21.

$$b=\frac{21\sqrt{7}}{21}$$

Cancel out 21 and 21.

$$b=\sqrt{7}$$

The system is now solved.

$$a=\sqrt{3} b=\sqrt{7}$$