» » » Rationalize $$\cfrac{(4\sqrt{7} + \sqrt{2})}{(\sqrt{2} - 7)}$$.

# Rationalize $$\cfrac{(4\sqrt{7} + \sqrt{2})}{(\sqrt{2} - 7)}$$.

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### Question

Rationalize $$\cfrac{(4\sqrt{7} + \sqrt{2})}{(\sqrt{2} - 7)}$$.

### Options

A)
-√14 - 6 B)
2√14 - 6
C)
-3√14 - 28
D)
√14 - 8

### Explanation:

Rationalizing the denominator in a fraction means to eliminate the radical in the denominator. We do this by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial of the form $$a + b$$ is $$a - b$$. Hence, the conjugate of $$\sqrt{2}-7$$ is $$\sqrt{2}+7$$.

Here's how we rationalize the given fraction:

$\cfrac{(4\sqrt{7} + \sqrt{2})}{(\sqrt{2} - 7)} \times \cfrac{(\sqrt{2} + 7)}{(\sqrt{2} + 7)}$

The denominator becomes $$(\sqrt{2} + 7)(\sqrt{2} - 7)$$, which is a difference of squares and simplifies to $$2 - 49 = -47$$.

The numerator becomes $$(4\sqrt{7} + \sqrt{2})(\sqrt{2} + 7)$$. Expand it to get $$-28\sqrt{7} - 6\sqrt{2} + 8\sqrt{14} + 14$$. Notice that both terms in the numerator have a common factor of 2 which can be factored out to get $$2(-14\sqrt{7} - 3\sqrt{2} + 4\sqrt{14} + 7)$$.

Divide the numerator and the denominator by $$-47$$ to get $$-\cfrac{2(-14\sqrt{7} - 3\sqrt{2} + 4\sqrt{14} + 7)}{47}$$.

When simplified, the expression becomes $$-\sqrt{14} - 6$$, which corresponds to Option A.