Solve frac{sqrt{21}-sqrt{7}}{sqrt{3}-1}

$$\frac{ \sqrt{ 21 } - \sqrt{ 7 } }{ \sqrt{ 3 } -1 }$$

Evaluate

$$\sqrt{7}\approx 2.645751311$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{21}-\sqrt{7}}{\sqrt{3}-1} by multiplying numerator and denominator by \sqrt{3}+1.

$$\frac{\left(\sqrt{21}-\sqrt{7}\right)\left(\sqrt{3}+1\right)}{\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}$$

Consider \left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{21}-\sqrt{7}\right)\left(\sqrt{3}+1\right)}{\left(\sqrt{3}\right)^{2}-1^{2}}$$

Square \sqrt{3}. Square 1.

$$\frac{\left(\sqrt{21}-\sqrt{7}\right)\left(\sqrt{3}+1\right)}{3-1}$$

Subtract 1 from 3 to get 2.

$$\frac{\left(\sqrt{21}-\sqrt{7}\right)\left(\sqrt{3}+1\right)}{2}$$

Apply the distributive property by multiplying each term of \sqrt{21}-\sqrt{7} by each term of \sqrt{3}+1.

$$\frac{\sqrt{21}\sqrt{3}+\sqrt{21}-\sqrt{7}\sqrt{3}-\sqrt{7}}{2}$$

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

$$\frac{\sqrt{3}\sqrt{7}\sqrt{3}+\sqrt{21}-\sqrt{7}\sqrt{3}-\sqrt{7}}{2}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{3\sqrt{7}+\sqrt{21}-\sqrt{7}\sqrt{3}-\sqrt{7}}{2}$$

To multiply \sqrt{7} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{3\sqrt{7}+\sqrt{21}-\sqrt{21}-\sqrt{7}}{2}$$

Combine \sqrt{21} and -\sqrt{21} to get 0.

$$\frac{3\sqrt{7}-\sqrt{7}}{2}$$

Combine 3\sqrt{7} and -\sqrt{7} to get 2\sqrt{7}.

$$\frac{2\sqrt{7}}{2}$$

Cancel out 2 and 2.

$$\sqrt{7}$$