Solve sqrt{(25-7)}sqrt{8100/314}

$$\sqrt { ( 25 - 7 ) } \sqrt { \frac { 8100 } { 314 } }$$

Evaluate

$$\frac{270\sqrt{157}}{157}\approx 21.548345881$$

Solution Steps

Subtract 7 from 25 to get 18.

$$\sqrt{18}\sqrt{\frac{8100}{314}}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$3\sqrt{2}\sqrt{\frac{8100}{314}}$$

Reduce the fraction \frac{8100}{314} to lowest terms by extracting and canceling out 2.

$$3\sqrt{2}\sqrt{\frac{4050}{157}}$$

Rewrite the square root of the division \sqrt{\frac{4050}{157}} as the division of square roots \frac{\sqrt{4050}}{\sqrt{157}}.

$$3\sqrt{2}\times \frac{\sqrt{4050}}{\sqrt{157}}$$

Factor 4050=45^{2}\times 2. Rewrite the square root of the product \sqrt{45^{2}\times 2} as the product of square roots \sqrt{45^{2}}\sqrt{2}. Take the square root of 45^{2}.

$$3\sqrt{2}\times \frac{45\sqrt{2}}{\sqrt{157}}$$

Rationalize the denominator of \frac{45\sqrt{2}}{\sqrt{157}} by multiplying numerator and denominator by \sqrt{157}.

$$3\sqrt{2}\times \frac{45\sqrt{2}\sqrt{157}}{\left(\sqrt{157}\right)^{2}}$$

The square of \sqrt{157} is 157.

$$3\sqrt{2}\times \frac{45\sqrt{2}\sqrt{157}}{157}$$

To multiply \sqrt{2} and \sqrt{157}, multiply the numbers under the square root.

$$3\sqrt{2}\times \frac{45\sqrt{314}}{157}$$

Express 3\times \frac{45\sqrt{314}}{157} as a single fraction.

$$\frac{3\times 45\sqrt{314}}{157}\sqrt{2}$$

Multiply 3 and 45 to get 135.

$$\frac{135\sqrt{314}}{157}\sqrt{2}$$

Express \frac{135\sqrt{314}}{157}\sqrt{2} as a single fraction.

$$\frac{135\sqrt{314}\sqrt{2}}{157}$$

Factor 314=2\times 157. Rewrite the square root of the product \sqrt{2\times 157} as the product of square roots \sqrt{2}\sqrt{157}.

$$\frac{135\sqrt{2}\sqrt{157}\sqrt{2}}{157}$$

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

$$\frac{135\times 2\sqrt{157}}{157}$$

Multiply 135 and 2 to get 270.

$$\frac{270\sqrt{157}}{157}$$