Solve 2.1/sqrt{1+2sqrt{frac{1{4}}}}

$$\frac{ 2.1 }{ \sqrt{ 1+2 \sqrt{ \frac{ 1 }{ 4 } } } }$$

Evaluate

$$\frac{21\sqrt{2}}{20}\approx 1.48492424$$

Solution Steps

Rewrite the square root of the division \frac{1}{4} as the division of square roots \frac{\sqrt{1}}{\sqrt{4}}. Take the square root of both numerator and denominator.

$$\frac{2.1}{\sqrt{1+2\times \frac{1}{2}}}$$

Cancel out 2 and 2.

$$\frac{2.1}{\sqrt{1+1}}$$

Add 1 and 1 to get 2.

$$\frac{2.1}{\sqrt{2}}$$

Rationalize the denominator of \frac{2.1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{2.1\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2.1\sqrt{2}}{2}$$

Divide 2.1\sqrt{2} by 2 to get 1.05\sqrt{2}.

$$1.05\sqrt{2}$$