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Roots and Exponents 700 Level Manhattan

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Intern
Joined: 21 Mar 2018
Posts: 6
Roots and Exponents 700 Level Manhattan  [#permalink]

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30 Nov 2019, 02:49
00:00

Difficulty:

35% (medium)

Question Stats:

80% (02:04) correct 20% (01:45) wrong based on 5 sessions

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If the sum of the cubes of a and b is 8 and a^6 – b^6 = 14, what is the value of a^3?

(A) 7/4

(B) 23/6

(C) 39/8

(D) 6

(E) 22/3

We know that the sum of the cubes of a and b is 8: a^3 + b^3 = 8. We also know that a^6 – b^6 = 14. Using our knowledge of the quadratic template for the difference of two squares,
x^2 – y^2 = (x + y)(x – y), we can rewrite a^6 – b^6 = 14 as follows:

(a^3)^2 – (b^3)^2 = 14
(a^3 – b^3)(a^3 + b^3) = 14

Substituting for a^3 + b^3 gives:

(a^3 – b^3)(8) = 14
(a^3 – b^3) = 14/8 = 7/4

Now that we have a value for a^3 + b^3 and value for a^3 – b^3, we can solve for the value of a^3 using elimination.

a^3 + b^3 = 8
a^3 – b^3 = 7/4

2a^3 = 39/4

Divide both sides by 2, and a^3 = 39/8.

Math Expert
Joined: 02 Sep 2009
Posts: 59622
Re: Roots and Exponents 700 Level Manhattan  [#permalink]

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30 Nov 2019, 03:08
1
1
YashKothari94 wrote:
If the sum of the cubes of a and b is 8 and a^6 – b^6 = 14, what is the value of a^3?

(A) 7/4

(B) 23/6

(C) 39/8

(D) 6

(E) 22/3

We know that the sum of the cubes of a and b is 8: a^3 + b^3 = 8. We also know that a^6 – b^6 = 14. Using our knowledge of the quadratic template for the difference of two squares,
x^2 – y^2 = (x + y)(x – y), we can rewrite a^6 – b^6 = 14 as follows:

(a^3)^2 – (b^3)^2 = 14
(a^3 – b^3)(a^3 + b^3) = 14

Substituting for a^3 + b^3 gives:

(a^3 – b^3)(8) = 14
(a^3 – b^3) = 14/8 = 7/4

Now that we have a value for a^3 + b^3 and value for a^3 – b^3, we can solve for the value of a^3 using elimination.

a^3 + b^3 = 8
a^3 – b^3 = 7/4

2a^3 = 39/4

Divide both sides by 2, and a^3 = 39/8.

Discussed here: https://gmatclub.com/forum/if-the-sum-o ... 88093.html
_________________
Re: Roots and Exponents 700 Level Manhattan   [#permalink] 30 Nov 2019, 03:08
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