Solve sqrt{{left(sqrt{frac{3/5}}{sqrt{7/5^2}}right)}^2}

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

Evaluate

$$\frac{\sqrt{105}}{7}\approx 1.463850109$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{3}{5}} as the division of square roots \frac{\sqrt{3}}{\sqrt{5}}.

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

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

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

The square of \sqrt{5} is 5.

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

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

$$\sqrt{\left(\frac{\frac{\sqrt{15}}{5}}{\sqrt{\frac{7}{5^{2}}}}\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\sqrt{\left(\frac{\frac{\sqrt{15}}{5}}{\sqrt{\frac{7}{25}}}\right)^{2}}$$

Rewrite the square root of the division \sqrt{\frac{7}{25}} as the division of square roots \frac{\sqrt{7}}{\sqrt{25}}.

$$\sqrt{\left(\frac{\frac{\sqrt{15}}{5}}{\frac{\sqrt{7}}{\sqrt{25}}}\right)^{2}}$$

Calculate the square root of 25 and get 5.

$$\sqrt{\left(\frac{\frac{\sqrt{15}}{5}}{\frac{\sqrt{7}}{5}}\right)^{2}}$$

Divide \frac{\sqrt{15}}{5} by \frac{\sqrt{7}}{5} by multiplying \frac{\sqrt{15}}{5} by the reciprocal of \frac{\sqrt{7}}{5}.

$$\sqrt{\left(\frac{\sqrt{15}\times 5}{5\sqrt{7}}\right)^{2}}$$

Cancel out 5 in both numerator and denominator.

$$\sqrt{\left(\frac{\sqrt{15}}{\sqrt{7}}\right)^{2}}$$

Rationalize the denominator of \frac{\sqrt{15}}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\sqrt{\left(\frac{\sqrt{15}\sqrt{7}}{\left(\sqrt{7}\right)^{2}}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\sqrt{\left(\frac{\sqrt{15}\sqrt{7}}{7}\right)^{2}}$$

To multiply \sqrt{15} and \sqrt{7}, multiply the numbers under the square root.

$$\sqrt{\left(\frac{\sqrt{105}}{7}\right)^{2}}$$

To raise \frac{\sqrt{105}}{7} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{\frac{\left(\sqrt{105}\right)^{2}}{7^{2}}}$$

The square of \sqrt{105} is 105.

$$\sqrt{\frac{105}{7^{2}}}$$

Calculate 7 to the power of 2 and get 49.

$$\sqrt{\frac{105}{49}}$$

Reduce the fraction \frac{105}{49} to lowest terms by extracting and canceling out 7.

$$\sqrt{\frac{15}{7}}$$

Rewrite the square root of the division \sqrt{\frac{15}{7}} as the division of square roots \frac{\sqrt{15}}{\sqrt{7}}.

$$\frac{\sqrt{15}}{\sqrt{7}}$$

Rationalize the denominator of \frac{\sqrt{15}}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{\sqrt{15}\sqrt{7}}{\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{\sqrt{15}\sqrt{7}}{7}$$

To multiply \sqrt{15} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\sqrt{105}}{7}$$