Solve frac{sqrt{21}-sqrt{7}}{sqrt{7}}

$$\frac{ \sqrt{ 21 } - \sqrt{ 7 } }{ \sqrt{ 7 } }$$

Evaluate

$$\sqrt{3}-1\approx 0.732050808$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{21}-\sqrt{7}}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{\left(\sqrt{21}-\sqrt{7}\right)\sqrt{7}}{\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{\left(\sqrt{21}-\sqrt{7}\right)\sqrt{7}}{7}$$

Use the distributive property to multiply \sqrt{21}-\sqrt{7} by \sqrt{7}.

$$\frac{\sqrt{21}\sqrt{7}-\left(\sqrt{7}\right)^{2}}{7}$$

Factor 21=7\times 3. Rewrite the square root of the product \sqrt{7\times 3} as the product of square roots \sqrt{7}\sqrt{3}.

$$\frac{\sqrt{7}\sqrt{3}\sqrt{7}-\left(\sqrt{7}\right)^{2}}{7}$$

Multiply \sqrt{7} and \sqrt{7} to get 7.

$$\frac{7\sqrt{3}-\left(\sqrt{7}\right)^{2}}{7}$$

The square of \sqrt{7} is 7.

$$\frac{7\sqrt{3}-7}{7}$$

Divide each term of 7\sqrt{3}-7 by 7 to get \sqrt{3}-1.

$$\sqrt{3}-1$$