Solve sqrt{1-(frac{3sqrt{105}}{35})^2}

$$\sqrt { 1 - ( \frac { 3 \sqrt { 105 } } { 35 } ) ^ { 2 } }$$

Evaluate

$$\frac{2\sqrt{70}}{35}\approx 0.478091444$$

Solution Steps

To raise \frac{3\sqrt{105}}{35} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{1-\frac{\left(3\sqrt{105}\right)^{2}}{35^{2}}}$$

Expand \left(3\sqrt{105}\right)^{2}.

$$\sqrt{1-\frac{3^{2}\left(\sqrt{105}\right)^{2}}{35^{2}}}$$

Calculate 3 to the power of 2 and get 9.

$$\sqrt{1-\frac{9\left(\sqrt{105}\right)^{2}}{35^{2}}}$$

The square of \sqrt{105} is 105.

$$\sqrt{1-\frac{9\times 105}{35^{2}}}$$

Multiply 9 and 105 to get 945.

$$\sqrt{1-\frac{945}{35^{2}}}$$

Calculate 35 to the power of 2 and get 1225.

$$\sqrt{1-\frac{945}{1225}}$$

Reduce the fraction \frac{945}{1225} to lowest terms by extracting and canceling out 35.

$$\sqrt{1-\frac{27}{35}}$$

Subtract \frac{27}{35} from 1 to get \frac{8}{35}.

$$\sqrt{\frac{8}{35}}$$

Rewrite the square root of the division \sqrt{\frac{8}{35}} as the division of square roots \frac{\sqrt{8}}{\sqrt{35}}.

$$\frac{\sqrt{8}}{\sqrt{35}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{2\sqrt{2}}{\sqrt{35}}$$

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

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

The square of \sqrt{35} is 35.

$$\frac{2\sqrt{2}\sqrt{35}}{35}$$

To multiply \sqrt{2} and \sqrt{35}, multiply the numbers under the square root.

$$\frac{2\sqrt{70}}{35}$$