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M70-21

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Joined: 02 Sep 2009
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03 Sep 2018, 04:16
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Which of the following equals $$(1 + \sqrt{\frac{5}{4}}+ \sqrt{\frac{3}{4}})^2 - (\sqrt{\frac{5}{4}}+ \sqrt{\frac{3}{4}})^2$$

A. $$\sqrt{5}+\sqrt{3}+1$$
B. $$1-\sqrt{3}-\sqrt{5}$$
C. $$2\sqrt{2}$$
D. $$4$$
E. $$4\sqrt{5}-4\sqrt{3}$$

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Math Expert
Joined: 02 Sep 2009
Posts: 52294

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03 Sep 2018, 04:16
Official Solution:

Which of the following equals $$(1 + \sqrt{\frac{5}{4}}+ \sqrt{\frac{3}{4}})^2 - (\sqrt{\frac{5}{4}}+ \sqrt{\frac{3}{4}})^2$$

A. $$\sqrt{5}+\sqrt{3}+1$$
B. $$1-\sqrt{3}-\sqrt{5}$$
C. $$2\sqrt{2}$$
D. $$4$$
E. $$4\sqrt{5}-4\sqrt{3}$$

The expression $$(1 + \sqrt{\frac{5}{4}}+ \sqrt{\frac{3}{4}})^2 - (\sqrt{\frac{5}{4}}+ \sqrt{\frac{3}{4}})^2$$ is of the form of the identity $$a^2 – b^2$$, that equals $$(a + b)(a – b)$$. We’ll use it for simplification:

$$(1 + \sqrt{\frac{5}{4}}+ \sqrt{\frac{3}{4}}+\sqrt{\frac{5}{4}}+ \sqrt{\frac{3}{4}})(1 + \sqrt{\frac{5}{4}}+ \sqrt{\frac{3}{4}}-\sqrt{\frac{5}{4}} - \sqrt{\frac{3}{4}})=$$

$$=(1 + 2*\sqrt{\frac{5}{4}}+ 2*\sqrt{\frac{3}{4}})(1)$$

Looking at the answers, we see none have fractions, so we’ll simplify:

$$=1 + 2*\sqrt{\frac{5}{4}}+ 2*\sqrt{\frac{3}{4}}$$

$$=1 + 2(\sqrt{\frac{5}{4}}+ \sqrt{\frac{3}{4}})$$

$$=1 + 2(\sqrt{\frac{1}{4}})(\sqrt{5}+ \sqrt{3})$$

$$=1 + 2(\frac{1}{2})(\sqrt{5}+ \sqrt{3})$$

$$=1 + \sqrt{5}+ \sqrt{3}$$

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Re M70-21 &nbs [#permalink] 03 Sep 2018, 04:16
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