Solve 7/1-2sqrt{2}+2+2sqrt{2}

$$\frac{ 7 }{ 1-2 \sqrt{ 2 } } +2+2 \sqrt{ 2 }$$

Evaluate

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Solution Steps

Rationalize the denominator of \frac{7}{1-2\sqrt{2}} by multiplying numerator and denominator by 1+2\sqrt{2}.

$$\frac{7\left(1+2\sqrt{2}\right)}{\left(1-2\sqrt{2}\right)\left(1+2\sqrt{2}\right)}+2+2\sqrt{2}$$

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

$$\frac{7\left(1+2\sqrt{2}\right)}{1^{2}-\left(-2\sqrt{2}\right)^{2}}+2+2\sqrt{2}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{7\left(1+2\sqrt{2}\right)}{1-\left(-2\sqrt{2}\right)^{2}}+2+2\sqrt{2}$$

Expand \left(-2\sqrt{2}\right)^{2}.

$$\frac{7\left(1+2\sqrt{2}\right)}{1-\left(-2\right)^{2}\left(\sqrt{2}\right)^{2}}+2+2\sqrt{2}$$

Calculate -2 to the power of 2 and get 4.

$$\frac{7\left(1+2\sqrt{2}\right)}{1-4\left(\sqrt{2}\right)^{2}}+2+2\sqrt{2}$$

The square of \sqrt{2} is 2.

$$\frac{7\left(1+2\sqrt{2}\right)}{1-4\times 2}+2+2\sqrt{2}$$

Multiply 4 and 2 to get 8.

$$\frac{7\left(1+2\sqrt{2}\right)}{1-8}+2+2\sqrt{2}$$

Subtract 8 from 1 to get -7.

$$\frac{7\left(1+2\sqrt{2}\right)}{-7}+2+2\sqrt{2}$$

Cancel out -7 and -7.

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

To find the opposite of 1+2\sqrt{2}, find the opposite of each term.

$$-1-2\sqrt{2}+2+2\sqrt{2}$$

Add -1 and 2 to get 1.

$$1-2\sqrt{2}+2\sqrt{2}$$

Combine -2\sqrt{2} and 2\sqrt{2} to get 0.

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Factor

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