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# If (a + 1/a)^2 = 3, find the value of a^3 + 1/a^3

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Math Expert
Joined: 02 Sep 2009
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If (a + 1/a)^2 = 3, find the value of a^3 + 1/a^3  [#permalink]

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04 Dec 2019, 01:41
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Difficulty:

55% (hard)

Question Stats:

58% (02:25) correct 42% (02:09) wrong based on 45 sessions

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If $$(a +\frac{1}{a})^2=3$$, find the value of $$a^3 + \frac{1}{a^3}$$

A. 0

B. 1

C. $$\sqrt{3}$$

D. $$2+\sqrt{3}$$

E. Not enough information

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Joined: 03 Nov 2019
Posts: 54
Re: If (a + 1/a)^2 = 3, find the value of a^3 + 1/a^3  [#permalink]

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04 Dec 2019, 02:09
1
using (a + b)^3 = a^3 + b^3 + 3ab(a + b)

a^3 + b^3 = (a + b)^3 - 3ab(a + b)

Now substituting the values from question:
a^3+(1/a)^3= (a+1/a)^3-3*a*1/a(a+1/a)
=$$\sqrt{3}$$^3-3*$$\sqrt{3}$$
=3$$\sqrt{3}$$-3$$\sqrt{3}$$
=0

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Joined: 26 Jun 2017
Posts: 21
Re: If (a + 1/a)^2 = 3, find the value of a^3 + 1/a^3  [#permalink]

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04 Dec 2019, 06:11
Given : $$(a+\frac{1}{a})^2 = 3$$

$$a+\frac{1}{a} = \sqrt{3}$$

$$(a+\frac{1}{a})^3$$ = 3$$\sqrt{3}$$

Lets just bother about LHS for now, Expanding LHS

$$a^3 + \frac{1}{a^3 }+ 3a^2\frac{1}{a} + 3a\frac{1}{a^2}$$

$$a^3 + \frac{1}{a^3 } + 3a + \frac{3}{a}$$

Taking 3 common,

$$a^3 + \frac{1}{a^3 } + 3(a + \frac{1}{a})$$

after substituting the value of $$a + \frac{1}{a }$$, the actual equation becomes:

$$a^3 + \frac{1}{a^3 } + 3\sqrt{3} = 3\sqrt{3}$$

$$a^3 + \frac{1}{a^3 } = 0$$

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Re: If (a + 1/a)^2 = 3, find the value of a^3 + 1/a^3  [#permalink]

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09 Dec 2019, 18:55
2
Bunuel wrote:
If $$(a +\frac{1}{a})^2=3$$, find the value of $$a^3 + \frac{1}{a^3}$$

A. 0

B. 1

C. $$\sqrt{3}$$

D. $$2+\sqrt{3}$$

E. Not enough information

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Since (a + 1/a)^2 = 3, a + 1/a = ±√3. Furthermore, since (a + 1/a)^2 = a^2 + 2a(1/a) + 1/a^2 = a^2 + 2 + 1/a^2 = 3, we see that a^2 + 1/a^2 = 1.

Now, if we multiply a + 1/a and a^2 + 1/a^2 (and assume that a + 1/a = √3), we have:

(a + 1/a)(a^2 + 1/a^2) = √3 x 1

a^3 + 1/a + a + 1/a^3 = √3

a^3 + √3 + 1/a^3 = √3

a^3 + 1/a^3 = 0

If a + 1/a = -√3, we have:

(a + 1/a)(a^2 + 1/a^2) = -√3 x 1

a^3 + 1/a + a + 1/a^3 = -√3

a^3 - √3 + 1/a^3 = -√3

a^3 + 1/a^3 = 0

We see that either way, a^3 + 1/a^3 = 0.

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Re: If (a + 1/a)^2 = 3, find the value of a^3 + 1/a^3   [#permalink] 09 Dec 2019, 18:55
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