Solve sqrt{{left(-4/15right)}^2-1{left(1/5right)}^2+{left(1/3right)}^2}

$$\sqrt{ { \left( \frac{ -4 }{ 15 } \right) }^{ 2 } -1 { \left( \frac{ 1 }{ 5 } \right) }^{ 2 } + { \left( \frac{ 1 }{ 3 } \right) }^{ 2 } }$$

Evaluate

$$\frac{4\sqrt{2}}{15}\approx 0.377123617$$

Solution Steps

Fraction \frac{-4}{15} can be rewritten as -\frac{4}{15} by extracting the negative sign.

$$\sqrt{\left(-\frac{4}{15}\right)^{2}-1\times \left(\frac{1}{5}\right)^{2}+\left(\frac{1}{3}\right)^{2}}$$

Calculate -\frac{4}{15} to the power of 2 and get \frac{16}{225}.

$$\sqrt{\frac{16}{225}-1\times \left(\frac{1}{5}\right)^{2}+\left(\frac{1}{3}\right)^{2}}$$

Calculate \frac{1}{5} to the power of 2 and get \frac{1}{25}.

$$\sqrt{\frac{16}{225}-1\times \frac{1}{25}+\left(\frac{1}{3}\right)^{2}}$$

Multiply 1 and \frac{1}{25} to get \frac{1}{25}.

$$\sqrt{\frac{16}{225}-\frac{1}{25}+\left(\frac{1}{3}\right)^{2}}$$

Subtract \frac{1}{25} from \frac{16}{225} to get \frac{7}{225}.

$$\sqrt{\frac{7}{225}+\left(\frac{1}{3}\right)^{2}}$$

Calculate \frac{1}{3} to the power of 2 and get \frac{1}{9}.

$$\sqrt{\frac{7}{225}+\frac{1}{9}}$$

Add \frac{7}{225} and \frac{1}{9} to get \frac{32}{225}.

$$\sqrt{\frac{32}{225}}$$

Rewrite the square root of the division \sqrt{\frac{32}{225}} as the division of square roots \frac{\sqrt{32}}{\sqrt{225}}.

$$\frac{\sqrt{32}}{\sqrt{225}}$$

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}.

$$\frac{4\sqrt{2}}{\sqrt{225}}$$

Calculate the square root of 225 and get 15.

$$\frac{4\sqrt{2}}{15}$$