Solve sqrt{32}-1/35sqrt{392}+1/240sqrt{97^2-47^2}=

$$\sqrt{ 32 } - \frac{ 1 }{ 35 } \sqrt{ 392 } + \frac{ 1 }{ 240 } \sqrt{ { 97 }^{ 2 } - { 47 }^{ 2 } } =$$

Evaluate

$$\frac{77\sqrt{2}}{20}\approx 5.444722215$$

Solution Steps

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$4\sqrt{2}-\frac{1}{35}\sqrt{392}+\frac{1}{240}\sqrt{97^{2}-47^{2}}$$

Factor 392=14^{2}\times 2. Rewrite the square root of the product \sqrt{14^{2}\times 2} as the product of square roots \sqrt{14^{2}}\sqrt{2}. Take the square root of 14^{2}.

$$4\sqrt{2}-\frac{1}{35}\times 14\sqrt{2}+\frac{1}{240}\sqrt{97^{2}-47^{2}}$$

Express -\frac{1}{35}\times 14 as a single fraction.

$$4\sqrt{2}+\frac{-14}{35}\sqrt{2}+\frac{1}{240}\sqrt{97^{2}-47^{2}}$$

Reduce the fraction \frac{-14}{35} to lowest terms by extracting and canceling out 7.

$$4\sqrt{2}-\frac{2}{5}\sqrt{2}+\frac{1}{240}\sqrt{97^{2}-47^{2}}$$

Combine 4\sqrt{2} and -\frac{2}{5}\sqrt{2} to get \frac{18}{5}\sqrt{2}.

$$\frac{18}{5}\sqrt{2}+\frac{1}{240}\sqrt{97^{2}-47^{2}}$$

Calculate 97 to the power of 2 and get 9409.

$$\frac{18}{5}\sqrt{2}+\frac{1}{240}\sqrt{9409-47^{2}}$$

Calculate 47 to the power of 2 and get 2209.

$$\frac{18}{5}\sqrt{2}+\frac{1}{240}\sqrt{9409-2209}$$

Subtract 2209 from 9409 to get 7200.

$$\frac{18}{5}\sqrt{2}+\frac{1}{240}\sqrt{7200}$$

Factor 7200=60^{2}\times 2. Rewrite the square root of the product \sqrt{60^{2}\times 2} as the product of square roots \sqrt{60^{2}}\sqrt{2}. Take the square root of 60^{2}.

$$\frac{18}{5}\sqrt{2}+\frac{1}{240}\times 60\sqrt{2}$$

Multiply \frac{1}{240} and 60 to get \frac{60}{240}.

$$\frac{18}{5}\sqrt{2}+\frac{60}{240}\sqrt{2}$$

Reduce the fraction \frac{60}{240} to lowest terms by extracting and canceling out 60.

$$\frac{18}{5}\sqrt{2}+\frac{1}{4}\sqrt{2}$$

Combine \frac{18}{5}\sqrt{2} and \frac{1}{4}\sqrt{2} to get \frac{77}{20}\sqrt{2}.

$$\frac{77}{20}\sqrt{2}$$