Solve sqrt{1+1/3^2+1/4^2}

$$\sqrt{ 1+ \frac{ 1 }{ { 3 }^{ 2 } } + \frac{ 1 }{ { 4 }^{ 2 } } }$$

Evaluate

$$\frac{13}{12}\approx 1.083333333$$

Solution Steps

Calculate 3 to the power of 2 and get 9.

$$\sqrt{1+\frac{1}{9}+\frac{1}{4^{2}}}$$

Convert 1 to fraction \frac{9}{9}.

$$\sqrt{\frac{9}{9}+\frac{1}{9}+\frac{1}{4^{2}}}$$

Since \frac{9}{9} and \frac{1}{9} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{9+1}{9}+\frac{1}{4^{2}}}$$

Add 9 and 1 to get 10.

$$\sqrt{\frac{10}{9}+\frac{1}{4^{2}}}$$

Calculate 4 to the power of 2 and get 16.

$$\sqrt{\frac{10}{9}+\frac{1}{16}}$$

Least common multiple of 9 and 16 is 144. Convert \frac{10}{9} and \frac{1}{16} to fractions with denominator 144.

$$\sqrt{\frac{160}{144}+\frac{9}{144}}$$

Since \frac{160}{144} and \frac{9}{144} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{160+9}{144}}$$

Add 160 and 9 to get 169.

$$\sqrt{\frac{169}{144}}$$

Rewrite the square root of the division \frac{169}{144} as the division of square roots \frac{\sqrt{169}}{\sqrt{144}}. Take the square root of both numerator and denominator.

$$\frac{13}{12}$$

Factor

$$\frac{13}{2 ^ {2} \cdot 3} = 1\frac{1}{12} = 1.0833333333333333$$