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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
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 \(\,{\left( {{r^3} + {1 \over {{r^3}}}} \right)^2}\,\) ?
\(\eqalign{\\
& \left( A \right)\,\,7.75 \cr \\
& \left( B \right)\,\,10 \cr \\
& \left( C \right)\,\,12.25 \cr \\
& \left( D \right)\,\,15 \cr \\
& \left( E \right)\,\,20 \cr}\)
If \({\left( {r + {1 \over r}} \right)^2} = 5\), what is the value of \(\,{\left( {{r^3} + {1 \over {{r^3}}}} \right)^2}\,\) ?
\(\eqalign{\\
& \left( A \right)\,\,7.75 \cr \\
& \left( B \right)\,\,10 \cr \\
& \left( C \right)\,\,12.25 \cr \\
& \left( D \right)\,\,15 \cr \\
& \left( E \right)\,\,20 \cr}\)
Tutor
Joined: 04 Aug 2010
Posts: 1333
Given Kudos: 9
Schools:Dartmouth College
If (r+1/r)^2 = 5, what is the value of (r^3+1/r^3)^2?
[#permalink]
Updated on: 21 Mar 2019, 17:53
Updated on: 21 Mar 2019, 17:53
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fskilnik
GMATH practice exercise (Quant Class 3)
If \({\left( {r + {1 \over r}} \right)^2} = 5\), what is the value of \(\,{\left( {{r^3} + {1 \over {{r^3}}}} \right)^2}\,\) ?
\(\eqalign{\\
& \left( A \right)\,\,7.75 \cr \\
& \left( B \right)\,\,10 \cr \\
& \left( C \right)\,\,12.25 \cr \\
& \left( D \right)\,\,15 \cr \\
& \left( E \right)\,\,20 \cr}\)
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 = 3
Multiplying 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√5
Substituting 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
.
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.
Last edited by GMATGuruNY on 21 Mar 2019, 17:53, edited 1 time in total.
Re: If (r+1/r)^2 = 5, what is the value of (r^3+1/r^3)^2?
[#permalink]
21 Mar 2019, 14:23
21 Mar 2019, 14:23
1
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Expert Reply
fskilnik
GMATH practice exercise (Quant Class 3)
If \({\left( {r + {1 \over r}} \right)^2} = 5\), what is the value of \(\,{\left( {{r^3} + {1 \over {{r^3}}}} \right)^2}\,\) ?
\(\eqalign{\\
& \left( A \right)\,\,7.75 \cr \\
& \left( B \right)\,\,10 \cr \\
& \left( C \right)\,\,12.25 \cr \\
& \left( D \right)\,\,15 \cr \\
& \left( E \right)\,\,20 \cr}\)
\({\left( {r + {1 \over r}} \right)^2} = 5\,\,\,\,\,\left( * \right)\)If \({\left( {r + {1 \over r}} \right)^2} = 5\), what is the value of \(\,{\left( {{r^3} + {1 \over {{r^3}}}} \right)^2}\,\) ?
\(\eqalign{\\
& \left( A \right)\,\,7.75 \cr \\
& \left( B \right)\,\,10 \cr \\
& \left( C \right)\,\,12.25 \cr \\
& \left( D \right)\,\,15 \cr \\
& \left( E \right)\,\,20 \cr}\)
\(? = \,{\left( {{r^3} + {1 \over {{r^3}}}} \right)^2}\)
\(?\,\,\, = \,\,\,{\left[ {\left( {r + {1 \over r}} \right)\left( {{r^2} - r \cdot {1 \over r} + {1 \over {{r^2}}}} \right)} \right]^{\,2}} = {\left( {r + {1 \over r}} \right)^2}{\left( {{r^2} - 1 + {1 \over {{r^2}}}} \right)^2}\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,5 \cdot {\left( {{r^2} - 1 + {1 \over {{r^2}}}} \right)^2}\,\,\,\mathop = \limits^{\left( {**} \right)} \,\,\,20\)
\(\left( {**} \right)\,\,{r^2} - 1 + {1 \over {{r^2}}} = \left( {{r^2} + 2 + {1 \over {{r^2}}}} \right) - 3 = {\left( {r + {1 \over r}} \right)^2} - 3\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,2\)
The correct answer is (E).
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
