Solve frac{3sqrt{2}+sqrt{5}}{sqrt{2}+1}

$$\frac{ 3 \sqrt{ 2 } + \sqrt{ 5 } }{ \sqrt{ 2 } +1 }$$

Evaluate

$$\sqrt{10}+6-\sqrt{5}-3\sqrt{2}\approx 2.683568996$$

Solution Steps

Rationalize the denominator of \frac{3\sqrt{2}+\sqrt{5}}{\sqrt{2}+1} by multiplying numerator and denominator by \sqrt{2}-1.

$$\frac{\left(3\sqrt{2}+\sqrt{5}\right)\left(\sqrt{2}-1\right)}{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}$$

Consider \left(\sqrt{2}+1\right)\left(\sqrt{2}-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}.

$$\frac{\left(3\sqrt{2}+\sqrt{5}\right)\left(\sqrt{2}-1\right)}{\left(\sqrt{2}\right)^{2}-1^{2}}$$

Square \sqrt{2}. Square 1.

$$\frac{\left(3\sqrt{2}+\sqrt{5}\right)\left(\sqrt{2}-1\right)}{2-1}$$

Subtract 1 from 2 to get 1.

$$\frac{\left(3\sqrt{2}+\sqrt{5}\right)\left(\sqrt{2}-1\right)}{1}$$

Anything divided by one gives itself.

$$\left(3\sqrt{2}+\sqrt{5}\right)\left(\sqrt{2}-1\right)$$

Apply the distributive property by multiplying each term of 3\sqrt{2}+\sqrt{5} by each term of \sqrt{2}-1.

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

The square of \sqrt{2} is 2.

$$3\times 2-3\sqrt{2}+\sqrt{5}\sqrt{2}-\sqrt{5}$$

Multiply 3 and 2 to get 6.

$$6-3\sqrt{2}+\sqrt{5}\sqrt{2}-\sqrt{5}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$6-3\sqrt{2}+\sqrt{10}-\sqrt{5}$$