Solve 1=frac{sqrt{sqrt{11}-3}sqrt{3+sqrt{11}}}{sqrt{8}}

$$1 = \frac{ \sqrt{ \sqrt{ 11 } -3 } \sqrt{ 3+ \sqrt{ 11 } } }{ \sqrt{ 8 } }$$

Verify

Short Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$1=\frac{\sqrt{\sqrt{11}-3}\sqrt{3+\sqrt{11}}}{2\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{\sqrt{11}-3}\sqrt{3+\sqrt{11}}}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$1=\frac{\sqrt{\sqrt{11}-3}\sqrt{3+\sqrt{11}}\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$1=\frac{\sqrt{\sqrt{11}-3}\sqrt{3+\sqrt{11}}\sqrt{2}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$1=\frac{\sqrt{\sqrt{11}-3}\sqrt{3+\sqrt{11}}\sqrt{2}}{4}$$

Subtract \frac{\sqrt{\sqrt{11}-3}\sqrt{3+\sqrt{11}}\sqrt{2}}{4} from both sides.

$$1-\frac{\sqrt{\sqrt{11}-3}\sqrt{3+\sqrt{11}}\sqrt{2}}{4}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{4}{4}.

$$\frac{4}{4}-\frac{\sqrt{\sqrt{11}-3}\sqrt{3+\sqrt{11}}\sqrt{2}}{4}=0$$

Since \frac{4}{4} and \frac{\sqrt{\sqrt{11}-3}\sqrt{3+\sqrt{11}}\sqrt{2}}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4-\sqrt{\sqrt{11}-3}\sqrt{3+\sqrt{11}}\sqrt{2}}{4}=0$$

Do the multiplications in 4-\sqrt{\sqrt{11}-3}\sqrt{3+\sqrt{11}}\sqrt{2}.

$$\frac{4-2}{4}=0$$

Do the calculations in 4-2.

$$\frac{2}{4}=0$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$\frac{1}{2}=0$$

Compare \frac{1}{2} and 0.

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