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# If x is a positive integer such that 4*2^(x+2.5)

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Intern
Joined: 18 Oct 2015
Posts: 8
If x is a positive integer such that 4*2^(x+2.5)  [#permalink]

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Updated on: 06 Jan 2016, 06:59
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1
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Difficulty:

45% (medium)

Question Stats:

70% (02:04) correct 30% (02:10) wrong based on 151 sessions

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If x is a positive integer such that $$4*2^{(x+2.5)} =\sqrt{2}*0.5^{(-3x)}$$. What is the value of $$x^3$$?

A) 2
B) 3
C) 5
D) 7
E) 8

Originally posted by timothyhenman1 on 05 Jan 2016, 21:21.
Last edited by Bunuel on 06 Jan 2016, 06:59, edited 1 time in total.
Renamed the topic and edited the question.
Senior Manager
Joined: 20 Aug 2015
Posts: 379
Location: India
GMAT 1: 760 Q50 V44
Re: If x is a positive integer such that 4*2^(x+2.5)  [#permalink]

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06 Jan 2016, 04:30
5
timothyhenman1 wrote:
If x is a positive integer such that 4*2^(x+2.5) = $$\sqrt{2}$$*0.5^(-3x)

What is the value of $$X^3$$?

A)2
B)3
C)5
D)7
E)8

This can be solved by bringing every term to the same base.
$$4*2^{x+2.5}$$ = $$\sqrt{2}$$*$$0.5^{-3x}$$

$$2^2*2^{x+2.5}$$ = $$2^{1/2}$$*$$(1/2)^{-3x}$$
$$2^{x +4.5}$$ =$$2^{1/2}$$*$$(2)^{3x}$$

$$2^{x +4.5}$$ =$$2^{3x + 0.5}$$
Equating the powers,

x + 4.5 = 3x + 0.5
2x = 4, x = 2

Hence, $$2^3$$ = 8 Option E
##### General Discussion
Intern
Joined: 18 Oct 2015
Posts: 8
Re: If x is a positive integer such that 4*2^(x+2.5)  [#permalink]

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06 Jan 2016, 10:36
TeamGMATIFY wrote:
timothyhenman1 wrote:
If x is a positive integer such that 4*2^(x+2.5) = $$\sqrt{2}$$*0.5^(-3x)

What is the value of $$X^3$$?

A)2
B)3
C)5
D)7
E)8

This can be solved by bringing every term to the same base.
$$4*2^{x+2.5}$$ = $$\sqrt{2}$$*$$0.5^{-3x}$$

$$2^2*2^{x+2.5}$$ = $$2^{1/2}$$*$$(1/2)^{-3x}$$
$$2^{x +4.5}$$ =$$2^{1/2}$$*$$(2)^{3x}$$

$$2^{x +4.5}$$ =$$2^{3x + 0.5}$$
Equating the powers,

x + 4.5 = 3x + 0.5
2x = 4, x = 2

Hence, $$2^3$$ = 8 Option E

Ugh! I feel so stupid, my first instinct was to change the base of the 4 to 2^2 but when it came to the second side of the eqn i my brain completely forgot that i can change root2 to 2^1/2 and .5^-3x completely intimidated me to the point where i ended up just guessing on the test.

thanks for your help in explaining this, it really helps. I will now go back and review those pesky exponent rules and tricks.
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Joined: 20 Aug 2015
Posts: 379
Location: India
GMAT 1: 760 Q50 V44
Re: If x is a positive integer such that 4*2^(x+2.5)  [#permalink]

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06 Jan 2016, 22:41
timothyhenman1 wrote:

Ugh! I feel so stupid, my first instinct was to change the base of the 4 to 2^2 but when it came to the second side of the eqn i my brain completely forgot that i can change root2 to 2^1/2 and .5^-3x completely intimidated me to the point where i ended up just guessing on the test.

thanks for your help in explaining this, it really helps. I will now go back and review those pesky exponent rules and tricks.

Always try to bring everything to the smallest possible base. This would ease out the calculations.
Intern
Joined: 05 Nov 2016
Posts: 2
If x is a positive integer such that 4*2^(x+2.5)  [#permalink]

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02 Dec 2016, 00:54
Question says x is a positive integer. the only possible choice where x^3 is an integer is e
Senior Manager
Joined: 27 Feb 2014
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GMAT 1: 570 Q49 V20
GPA: 3.97
WE: Engineering (Education)
Re: If x is a positive integer such that 4*2^(x+2.5)  [#permalink]

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14 Nov 2019, 00:23
timothyhenman1 wrote:
If x is a positive integer such that $$4*2^{(x+2.5)} =\sqrt{2}*0.5^{(-3x)}$$. What is the value of $$x^3$$?

A) 2
B) 3
C) 5
D) 7
E) 8

2^(x+4.5) = 2^1/2 * 2^3x
2^(x+4.5) = 2^(3x + 0.5)
Therefore, x+4.5 = 3x+0.5
x = 2
x^3 = 8

E is correct
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Re: If x is a positive integer such that 4*2^(x+2.5)  [#permalink]

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18 Nov 2019, 08:39
timothyhenman1 wrote:
If x is a positive integer such that $$4*2^{(x+2.5)} =\sqrt{2}*0.5^{(-3x)}$$. What is the value of $$x^3$$?

A) 2
B) 3
C) 5
D) 7
E) 8

Simplifying the equation, we have:

2^2 * 2^(x + 2.5) = 2^(1/2) * (1/2)^(-3x)

Noting that (1/2)^(-3x) is equivalent to 2^(3x), we have:

2^(x + 4.5) = 2^(0.5) * 2^(3x)

2^(x + 4.5) = 2^(0.5 + 3x)

When the bases are equal, we can equate the exponents:

x + 4.5 = 0.5 + 3x

4 = 2x

2 = x

Thus, x^3 = 2^3 = 8.

Alternate Solution:

Keeping in mind that x is a positive integer, x^3 must be a perfect cube. The only perfect cube among the given choices is 8.

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Re: If x is a positive integer such that 4*2^(x+2.5)   [#permalink] 18 Nov 2019, 08:39