Solve frac{4+2sqrt{2}}{sqrt{2}+1}

$$\frac { 4 + 2 \sqrt { 2 } } { \sqrt { 2 } + 1 }$$

Evaluate

$$2\sqrt{2}\approx 2.828427125$$

Solution Steps

Rationalize the denominator of \frac{4+2\sqrt{2}}{\sqrt{2}+1} by multiplying numerator and denominator by \sqrt{2}-1.

$$\frac{\left(4+2\sqrt{2}\right)\left(\sqrt{2}-1\right)}{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}$$

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}.

$$\frac{\left(4+2\sqrt{2}\right)\left(\sqrt{2}-1\right)}{\left(\sqrt{2}\right)^{2}-1^{2}}$$

Square \sqrt{2}. Square 1.

$$\frac{\left(4+2\sqrt{2}\right)\left(\sqrt{2}-1\right)}{2-1}$$

Subtract 1 from 2 to get 1.

$$\frac{\left(4+2\sqrt{2}\right)\left(\sqrt{2}-1\right)}{1}$$

Anything divided by one gives itself.

$$\left(4+2\sqrt{2}\right)\left(\sqrt{2}-1\right)$$

Apply the distributive property by multiplying each term of 4+2\sqrt{2} by each term of \sqrt{2}-1.

$$4\sqrt{2}-4+2\left(\sqrt{2}\right)^{2}-2\sqrt{2}$$

The square of \sqrt{2} is 2.

$$4\sqrt{2}-4+2\times 2-2\sqrt{2}$$

Multiply 2 and 2 to get 4.

$$4\sqrt{2}-4+4-2\sqrt{2}$$

Add -4 and 4 to get 0.

$$4\sqrt{2}-2\sqrt{2}$$

Combine 4\sqrt{2} and -2\sqrt{2} to get 2\sqrt{2}.

$$2\sqrt{2}$$