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GMATH Teacher
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If (r+1/r)^2 = 5, what is the value of (r^3+1/r^3)^2?
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21 Mar 2019, 12:50
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GMATH practice exercise (Quant Class 3) If \({\left( {r + {1 \over r}} \right)^2} = 5\), what is the value of \(\,{\left( {{r^3} + {1 \over {{r^3}}}} \right)^2}\,\) ? \(\eqalign{ & \left( A \right)\,\,7.75 \cr & \left( B \right)\,\,10 \cr & \left( C \right)\,\,12.25 \cr & \left( D \right)\,\,15 \cr & \left( E \right)\,\,20 \cr}\)
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Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our highlevel "quant" preparation starts here: https://gmath.net



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If (r+1/r)^2 = 5, what is the value of (r^3+1/r^3)^2?
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Updated on: 21 Mar 2019, 17:53
fskilnik wrote: GMATH practice exercise (Quant Class 3)
If \({\left( {r + {1 \over r}} \right)^2} = 5\), what is the value of \(\,{\left( {{r^3} + {1 \over {{r^3}}}} \right)^2}\,\) ?
\(\eqalign{ & \left( A \right)\,\,7.75 \cr & \left( B \right)\,\,10 \cr & \left( C \right)\,\,12.25 \cr & \left( D \right)\,\,15 \cr & \left( E \right)\,\,20 \cr}\) (r + 1/r)² = 5 r + 1/r = √5(r + 1/r)² = 5 r² + 1/r² + 2(r)(1/r) = 5 r² + 1/r² + 2 = 5 r² + 1/r² = 3Multiplying the blue equation and the red equation, we get: (r² + 1/r²)(r + 1/r) = 3√5 r³ + (r²)(1/r) + (1/r²)(r) + 1/r³ = 3√5 r³ + (r + 1/r) + 1/r³ = 3√5Substituting r + 1/r = √5 into the green equation above, we get: r³ + √5 + 1/r³ = 3√5 r³ + 1/r³ = 2√5 (r³ + 1/r³)² = (2√5)² (r³ + 1/r³)² = 20 .
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Originally posted by GMATGuruNY on 21 Mar 2019, 13:54.
Last edited by GMATGuruNY on 21 Mar 2019, 17:53, edited 1 time in total.



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Re: If (r+1/r)^2 = 5, what is the value of (r^3+1/r^3)^2?
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21 Mar 2019, 14:23
fskilnik wrote: GMATH practice exercise (Quant Class 3)
If \({\left( {r + {1 \over r}} \right)^2} = 5\), what is the value of \(\,{\left( {{r^3} + {1 \over {{r^3}}}} \right)^2}\,\) ?
\(\eqalign{ & \left( A \right)\,\,7.75 \cr & \left( B \right)\,\,10 \cr & \left( C \right)\,\,12.25 \cr & \left( D \right)\,\,15 \cr & \left( E \right)\,\,20 \cr}\)
\({\left( {r + {1 \over r}} \right)^2} = 5\,\,\,\,\,\left( * \right)\) \(? = \,{\left( {{r^3} + {1 \over {{r^3}}}} \right)^2}\) \(?\,\,\, = \,\,\,{\left[ {\left( {r + {1 \over r}} \right)\left( {{r^2}  r \cdot {1 \over r} + {1 \over {{r^2}}}} \right)} \right]^{\,2}} = {\left( {r + {1 \over r}} \right)^2}{\left( {{r^2}  1 + {1 \over {{r^2}}}} \right)^2}\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,5 \cdot {\left( {{r^2}  1 + {1 \over {{r^2}}}} \right)^2}\,\,\,\mathop = \limits^{\left( {**} \right)} \,\,\,20\) \(\left( {**} \right)\,\,{r^2}  1 + {1 \over {{r^2}}} = \left( {{r^2} + 2 + {1 \over {{r^2}}}} \right)  3 = {\left( {r + {1 \over r}} \right)^2}  3\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,2\) The correct answer is (E). We follow the notations and rationale taught in the GMATH method. Regards, Fabio.
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Re: If (r+1/r)^2 = 5, what is the value of (r^3+1/r^3)^2?
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21 Mar 2019, 14:32
I would go for E. Please show a quick way. Posted from my mobile device
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Re: If (r+1/r)^2 = 5, what is the value of (r^3+1/r^3)^2?
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26 Mar 2019, 20:15
It is pretty straight forward if dealt this way 
(r+1/r)^3 = r^3 +(1/r)^3 + 3(r*(1/r))(r+(1/r)) (r+1/r)^3 = r^3 +(1/r)^3 + 3(r+(1/r))
We know that r+(1/r) = (5)^1/2
So r^3 +(1/r)^3 = 2*(5^(1/2)) Square the ans = 20



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Re: If (r+1/r)^2 = 5, what is the value of (r^3+1/r^3)^2?
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28 Mar 2019, 10:35
GMATGuruNY wrote: (r + 1/r)² = 5 r + 1/r = √5
Hi GMATGuruNYshould not the highlighted part above be r + 1/r = √5 Why did you avoid modulus sign?



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Re: If (r+1/r)^2 = 5, what is the value of (r^3+1/r^3)^2?
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28 Mar 2019, 12:48
Mo2men wrote: GMATGuruNY wrote: (r + 1/r)² = 5 r + 1/r = √5
Hi GMATGuruNYshould not the highlighted part above be r + 1/r = √5 Why did you avoid modulus sign? Since \(r + \frac{1}{r} = √5\) yields one of the answer choices, there is no need to consider \(r + \frac{1}{r} = √5\).
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Re: If (r+1/r)^2 = 5, what is the value of (r^3+1/r^3)^2?
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