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1/(2-√3)^2=？

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by Max@Math Revolution » Fri Sep 21, 2018 12:18 am

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[Math Revolution GMAT math practice question]

1/(2-√3)^2=？

A. 2+√3
B. 2-√3
C. 7+4√3
D. 7-4√3
E. 4+7√3
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by fskilnik@GMATH » Fri Sep 21, 2018 6:19 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

1/(2-√3)^2=？

A. 2+√3
B. 2-√3
C. 7+4√3
D. 7-4√3
E. 4+7√3
Nice problem!
\[? = \frac{1}{{{{\left( {2 - \sqrt 3 } \right)}^2}}} = \frac{1}{{{{\left( {2 - \sqrt 3 } \right)}^2}}} \cdot \frac{{{{\left( {2 + \sqrt 3 } \right)}^2}}}{{{{\left( {2 + \sqrt 3 } \right)}^2}}}\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,{\left( {2 + \sqrt 3 } \right)^2} = 4 + 4\sqrt 3 + 3 = 7 + 4\sqrt 3 \]
\[\left( * \right)\,\,\,{\left( {2 - \sqrt 3 } \right)^2}{\left( {2 + \sqrt 3 } \right)^2} = {\left[ {\left( {2 - \sqrt 3 } \right)\left( {2 + \sqrt 3 } \right)} \right]^2} = {\left[ {{2^2} - {{\left( {\sqrt 3 } \right)}^2}} \right]^{\,2}} = 1\]

The correct answer is therefore [spoiler]__(C)____[/spoiler] .

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

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Jay@ManhattanReview » Sat Sep 22, 2018 8:27 pm
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

1/(2-√3)^2=？

A. 2+√3
B. 2-√3
C. 7+4√3
D. 7-4√3
E. 4+7√3
1/(2 - √3)^2
= 1/(4 + 3 - 4√3)
= 1/(7 - 4√3)

We see that there is no option as 1/(7 - 4√3), so let's rationalize the fraction.

Multiplying and dividing the fraction 1/(7 - 4√3) by (7 + 4√3), we get

[1/(7 - 4√3)] * [(7 + 4√3) / (7 + 4√3)]

(7 + 4√3) / [(7 - 4√3)(7 + 4√3)]

(7 + 4√3) / (49 - 48)

= 7 + 4√3

The correct answer: C

Hope this helps!

-Jay
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by Max@Math Revolution » Mon Sep 24, 2018 4:09 am
=>

1/(2-√3)^2= [1/(2-√3)]^2= {(2+√3)/{(2-√3)(2+√3)}^2={(2+√3)/(4-3)}^2=(2+√3)^2= 4+4√3+3 = 7+4√3.

Therefore, the answer is C.
Answer: C
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by Scott@TargetTestPrep » Thu Sep 27, 2018 4:22 pm
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

1/(2-√3)^2=？

A. 2+√3
B. 2-√3
C. 7+4√3
D. 7-4√3
E. 4+7√3
First, let's simplify the denominator: (2-√3)^2:

(2-√3)^2 = 2^2 - 2(2)(√3) + (√3)^2 = 4 - 4√3 + 3 = 7 - 4√3

We have a square root expression in the denominator, so we must rationalize the denominator. We do this by multiplying both the numerator and the denominator by the conjugate of 7 - 4√3, which is 7 + 4√3.

Now let's simplify 1/(7 - 4√3) by multiplying numerator and denominator by (7 + 4√3):

1/(7 - 4√3) x (7 + 4√3) / (7 + 4√3)

(7 + 4√3) / (7^2 - (4√3)^2)

(7 + 4√3)/(49 - 48)

7 + 4√3

Answer: C

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