Solve frac{10sqrt{2}}{sqrt{3}+3}

$$\frac{ 10 \sqrt{ 2 } }{ \sqrt{ 3 } +3 }$$

Evaluate

$$-\frac{5\sqrt{6}}{3}+5\sqrt{2}\approx 2.988584907$$

Solution Steps

Rationalize the denominator of \frac{10\sqrt{2}}{\sqrt{3}+3} by multiplying numerator and denominator by \sqrt{3}-3.

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

Consider \left(\sqrt{3}+3\right)\left(\sqrt{3}-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}.

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

Square \sqrt{3}. Square 3.

$$\frac{10\sqrt{2}\left(\sqrt{3}-3\right)}{3-9}$$

Subtract 9 from 3 to get -6.

$$\frac{10\sqrt{2}\left(\sqrt{3}-3\right)}{-6}$$

Use the distributive property to multiply 10\sqrt{2} by \sqrt{3}-3.

$$\frac{10\sqrt{2}\sqrt{3}-30\sqrt{2}}{-6}$$

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

$$\frac{10\sqrt{6}-30\sqrt{2}}{-6}$$