Solve frac{12+6sqrt{2}}{1+sqrt{2}}

$$\frac { 12 + 6 \sqrt { 2 } } { 1 + \sqrt { 2 } }$$

Evaluate

$$6\sqrt{2}\approx 8.485281374$$

Solution Steps

Rationalize the denominator of \frac{12+6\sqrt{2}}{1+\sqrt{2}} by multiplying numerator and denominator by 1-\sqrt{2}.

$$\frac{\left(12+6\sqrt{2}\right)\left(1-\sqrt{2}\right)}{\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)}$$

Consider \left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(12+6\sqrt{2}\right)\left(1-\sqrt{2}\right)}{1^{2}-\left(\sqrt{2}\right)^{2}}$$

Square 1. Square \sqrt{2}.

$$\frac{\left(12+6\sqrt{2}\right)\left(1-\sqrt{2}\right)}{1-2}$$

Subtract 2 from 1 to get -1.

$$\frac{\left(12+6\sqrt{2}\right)\left(1-\sqrt{2}\right)}{-1}$$

Anything divided by -1 gives its opposite.

$$-\left(12+6\sqrt{2}\right)\left(1-\sqrt{2}\right)$$

Apply the distributive property by multiplying each term of 12+6\sqrt{2} by each term of 1-\sqrt{2}.

$$-\left(12-12\sqrt{2}+6\sqrt{2}-6\left(\sqrt{2}\right)^{2}\right)$$

Combine -12\sqrt{2} and 6\sqrt{2} to get -6\sqrt{2}.

$$-\left(12-6\sqrt{2}-6\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$-\left(12-6\sqrt{2}-6\times 2\right)$$

Multiply -6 and 2 to get -12.

$$-\left(12-6\sqrt{2}-12\right)$$

Subtract 12 from 12 to get 0.

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

The opposite of -6\sqrt{2} is 6\sqrt{2}.

$$6\sqrt{2}$$