Solve frac{sqrt{3}-5}{sqrt{7}+5}

$$\frac { \sqrt { 3 } - 5 } { \sqrt { 7 } + 5 }$$

Evaluate

$$\frac{5\sqrt{3}+5\sqrt{7}-\sqrt{21}-25}{18}\approx -0.427420283$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{3}-5}{\sqrt{7}+5} by multiplying numerator and denominator by \sqrt{7}-5.

$$\frac{\left(\sqrt{3}-5\right)\left(\sqrt{7}-5\right)}{\left(\sqrt{7}+5\right)\left(\sqrt{7}-5\right)}$$

Consider \left(\sqrt{7}+5\right)\left(\sqrt{7}-5\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{3}-5\right)\left(\sqrt{7}-5\right)}{\left(\sqrt{7}\right)^{2}-5^{2}}$$

Square \sqrt{7}. Square 5.

$$\frac{\left(\sqrt{3}-5\right)\left(\sqrt{7}-5\right)}{7-25}$$

Subtract 25 from 7 to get -18.

$$\frac{\left(\sqrt{3}-5\right)\left(\sqrt{7}-5\right)}{-18}$$

Apply the distributive property by multiplying each term of \sqrt{3}-5 by each term of \sqrt{7}-5.

$$\frac{\sqrt{3}\sqrt{7}-5\sqrt{3}-5\sqrt{7}+25}{-18}$$

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

$$\frac{\sqrt{21}-5\sqrt{3}-5\sqrt{7}+25}{-18}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{21}+5\sqrt{3}+5\sqrt{7}-25}{18}$$