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(7+43+7-43)^2 is equal to which of the following?

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(7+43+7-43)^2 is equal to which of the following?

by Max@Math Revolution » Wed Jan 09, 2019 11:13 pm
$$\left(\sqrt{7+4\sqrt{3}}+\sqrt{7-4\sqrt{3}}\right)^2$$ is equal to which of the following?

A. 32
B. 30
C. 24
D. 16
E. 12
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by GMATGuruNY » Thu Jan 10, 2019 3:51 am
Max@Math Revolution wrote:$$\left(\sqrt{7+4\sqrt{3}}+\sqrt{7-4\sqrt{3}}\right)^2$$ is equal to which of the following?

A. 32
B. 30
C. 24
D. 16
E. 12
√3 ≈ 1.7.
7 + 4√3 ≈ 7 + (4)(1.7) = 13.8.
7 - 4√3 ≈ 7 - (4)(1.7) = 0.2.

Thus, the given expression can be approximated as follows:
(√13.8 + √0.2)² = a little more than 13.8.
Only D is viable.

The correct answer is D.
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by fskilnik@GMATH » Thu Jan 10, 2019 9:09 am
Max@Math Revolution wrote:$$\left(\sqrt{7+4\sqrt{3}}+\sqrt{7-4\sqrt{3}}\right)^2$$ is equal to which of the following?

A. 32
B. 30
C. 24
D. 16
E. 12
$$? = {\left( {\sqrt {7 + 4\sqrt 3 } + \sqrt {7 - 4\sqrt 3 } } \right)^2} = {\left( {A + B} \right)^2}$$
$${A^2} = 7 + 4\sqrt 3 $$
$${B^2} = 7 - 4\sqrt 3 $$
$$2AB = 2\sqrt {\left( {7 + 4\sqrt 3 } \right)\left( {7 - 4\sqrt 3 } \right)} = 2\sqrt {{7^2} - {{\left( {4\sqrt 3 } \right)}^2}} = 2\sqrt {49 - 48} = 2$$
$$? = \left( {7 + 4\sqrt 3 } \right) + \left( {7 - 4\sqrt 3 } \right) + 2 = 16$$

This solution follows the notations and rationale taught in the GMATH method.

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by Max@Math Revolution » Sun Jan 13, 2019 4:39 pm
=>

You should recall two properties about nested radicals:
$$\sqrt{a+b+2\sqrt{ab}}=\sqrt{a}+\sqrt{b}$$ ,
$$\sqrt{a+b-2\sqrt{ab}}=\sqrt{a}-\sqrt{b}$$ , where a > b.

Together, these yield
$$\left(\sqrt{7+4\sqrt{3}}+\sqrt{7-4\sqrt{3}}\right)^2$$
$$=\left(\sqrt{7+2\sqrt{12}}+\sqrt{7-2\sqrt{12}}\right)^2$$
$$=\left(\sqrt{4}+\sqrt{3}+\sqrt{4}-\sqrt{3}\right)^2$$
$$=\left(2\sqrt{4}\right)^2$$
$$=16$$

Therefore, the answer is D.
Answer: D
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by Scott@TargetTestPrep » Mon Jan 14, 2019 5:53 pm
Max@Math Revolution wrote:$$\left(\sqrt{7+4\sqrt{3}}+\sqrt{7-4\sqrt{3}}\right)^2$$ is equal to which of the following?

A. 32
B. 30
C. 24
D. 16
E. 12
We can look at the given expression as the quadratic identity of (x + y)^2 = x^2 + y^2 + 2xy, and thus:

x^2 = 7 + 4√3

y^2 = 7 - 4√3

Next we can determine the value of 2xy:

2(√(7 + 4√3)(√(7 - 4√3)

2√[(7 + 4√3)(7 - 4√3)]

Using the difference of squares, we have:

2√[7^2 - (4√3)^2]

2√(49 - 48) = 2

Thus, the final value is:

(7 + 4√3) + (7 - 4√3) + 2 = 16

Answer: D

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