Solve 1/sqrt{2}quadtext{(b)}quad1/sqrt{3}

$$\frac { 1 } { \sqrt { 2 } } \quad \text { (b) } \quad \frac { 1 } { \sqrt { 3 } }$$

Evaluate

$$\frac{\sqrt{6}b}{6}$$

Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}b\times \frac{1}{\sqrt{3}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}}{2}b\times \frac{1}{\sqrt{3}}$$

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\sqrt{2}}{2}b\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{2}}{2}b\times \frac{\sqrt{3}}{3}$$

Express \frac{\sqrt{2}}{2}b as a single fraction.

$$\frac{\sqrt{2}b}{2}\times \frac{\sqrt{3}}{3}$$

Multiply \frac{\sqrt{2}b}{2} times \frac{\sqrt{3}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{2}b\sqrt{3}}{2\times 3}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\sqrt{6}b}{2\times 3}$$

Multiply 2 and 3 to get 6.

$$\frac{\sqrt{6}b}{6}$$

Differentiate w.r.t. b

$$\frac{\sqrt{6}}{6} = 0.40824829046386296$$