If (r+1/r)^2 = 5, what is the value of (r^3+1/r^3)^2?

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fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Thu Mar 21, 2019 11:43 am

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GMATH practice exercise (Quant Class 3)

If (r+1/r)^2 = 5, what is the value of (r^3+1/r^3)^2?

(A) 7.75
(B) 10
(C) 12.25
(D) 15
(E) 20

Answer: [spoiler]_____(E)__[/spoiler]

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
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Source: Beat The GMAT — Problem Solving | Thu Mar 21, 2019 11:43 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Thu Mar 21, 2019 12:37 pm

fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 3)

If (r+1/r)^2 = 5, what is the value of (r^3+1/r^3)^2?

(A) 7.75
(B) 10
(C) 12.25
(D) 15
(E) 20

(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

The correct answer is E.

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fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

If (r+1/r)^2 = 5, what is the value of (r^3+1/r^3)^2?

by fskilnik@GMATH » Thu Mar 21, 2019 1:10 pm

fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 3)

If (r+1/r)^2 = 5, what is the value of (r^3+1/r^3)^2?

(A) 7.75
(B) 10
(C) 12.25
(D) 15
(E) 20

$${\left( {r + {1 \over r}} \right)^2} = 5\,\,\,\,\,\left( * \right)$$
$$? = {\left( {{r^3} + {1 \over {{r^3}}}} \right)^2}$$
$$?\,\,\, = \,\,\,{\left[ {\left( {r + {1 \over r}} \right)\left( {{r^2} - r \cdot {1 \over r} + {1 \over {{r^2}}}} \right)} \right]^{\,2}} = {\left( {r + {1 \over r}} \right)^2}{\left( {{r^2} - 1 + {1 \over {{r^2}}}} \right)^2}\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,5 \cdot {\left( {{r^2} - 1 + {1 \over {{r^2}}}} \right)^2}\,\,\,\mathop = \limits^{\left( {**} \right)} \,\,\,20$$
$$\left( {**} \right)\,\,{r^2} - 1 + {1 \over {{r^2}}} = \left( {{r^2} + 2 + {1 \over {{r^2}}}} \right) - 3 = {\left( {r + {1 \over r}} \right)^2} - 3\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,2$$

The correct answer is (E).

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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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