Solve frac{sqrt{7}}{sqrt{10}+3}

$$\frac { \sqrt { 7 } } { \sqrt { 10 } + 3 }$$

Evaluate

$$\sqrt{7}\left(\sqrt{10}-3\right)\approx 0.429346332$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{7}}{\sqrt{10}+3} by multiplying numerator and denominator by \sqrt{10}-3.

$$\frac{\sqrt{7}\left(\sqrt{10}-3\right)}{\left(\sqrt{10}+3\right)\left(\sqrt{10}-3\right)}$$

Consider \left(\sqrt{10}+3\right)\left(\sqrt{10}-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{7}\left(\sqrt{10}-3\right)}{\left(\sqrt{10}\right)^{2}-3^{2}}$$

Square \sqrt{10}. Square 3.

$$\frac{\sqrt{7}\left(\sqrt{10}-3\right)}{10-9}$$

Subtract 9 from 10 to get 1.

$$\frac{\sqrt{7}\left(\sqrt{10}-3\right)}{1}$$

Anything divided by one gives itself.

$$\sqrt{7}\left(\sqrt{10}-3\right)$$

Use the distributive property to multiply \sqrt{7} by \sqrt{10}-3.

$$\sqrt{7}\sqrt{10}-3\sqrt{7}$$

To multiply \sqrt{7} and \sqrt{10}, multiply the numbers under the square root.

$$\sqrt{70}-3\sqrt{7}$$