Solve sqrt{(15/28)^2+(15/14)^2+(45/28)^2}

$$\sqrt { ( \frac { 15 } { 28 } ) ^ { 2 } + ( \frac { 15 } { 14 } ) ^ { 2 } + ( \frac { 45 } { 28 } ) ^ { 2 } }$$

Evaluate

$$\frac{15\sqrt{14}}{28}\approx 2.004459314$$

Solution Steps

Calculate \frac{15}{28} to the power of 2 and get \frac{225}{784}.

$$\sqrt{\frac{225}{784}+\left(\frac{15}{14}\right)^{2}+\left(\frac{45}{28}\right)^{2}}$$

Calculate \frac{15}{14} to the power of 2 and get \frac{225}{196}.

$$\sqrt{\frac{225}{784}+\frac{225}{196}+\left(\frac{45}{28}\right)^{2}}$$

Least common multiple of 784 and 196 is 784. Convert \frac{225}{784} and \frac{225}{196} to fractions with denominator 784.

$$\sqrt{\frac{225}{784}+\frac{900}{784}+\left(\frac{45}{28}\right)^{2}}$$

Since \frac{225}{784} and \frac{900}{784} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{225+900}{784}+\left(\frac{45}{28}\right)^{2}}$$

Add 225 and 900 to get 1125.

$$\sqrt{\frac{1125}{784}+\left(\frac{45}{28}\right)^{2}}$$

Calculate \frac{45}{28} to the power of 2 and get \frac{2025}{784}.

$$\sqrt{\frac{1125}{784}+\frac{2025}{784}}$$

Since \frac{1125}{784} and \frac{2025}{784} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1125+2025}{784}}$$

Add 1125 and 2025 to get 3150.

$$\sqrt{\frac{3150}{784}}$$

Reduce the fraction \frac{3150}{784} to lowest terms by extracting and canceling out 14.

$$\sqrt{\frac{225}{56}}$$

Rewrite the square root of the division \sqrt{\frac{225}{56}} as the division of square roots \frac{\sqrt{225}}{\sqrt{56}}.

$$\frac{\sqrt{225}}{\sqrt{56}}$$

Calculate the square root of 225 and get 15.

$$\frac{15}{\sqrt{56}}$$

Factor 56=2^{2}\times 14. Rewrite the square root of the product \sqrt{2^{2}\times 14} as the product of square roots \sqrt{2^{2}}\sqrt{14}. Take the square root of 2^{2}.

$$\frac{15}{2\sqrt{14}}$$

Rationalize the denominator of \frac{15}{2\sqrt{14}} by multiplying numerator and denominator by \sqrt{14}.

$$\frac{15\sqrt{14}}{2\left(\sqrt{14}\right)^{2}}$$

The square of \sqrt{14} is 14.

$$\frac{15\sqrt{14}}{2\times 14}$$

Multiply 2 and 14 to get 28.

$$\frac{15\sqrt{14}}{28}$$