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

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timothyhenman1
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If x is a positive integer such that 4*2^(x+2.5) [#permalink] New post   Updated on: 06 Jan 2016, 07:59
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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
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E

Originally posted by timothyhenman1 on 05 Jan 2016, 22:21.
Last edited by Bunuel on 06 Jan 2016, 07:59, edited 1 time in total.
Renamed the topic and edited the question.
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TeamGMATIFY
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Re: If x is a positive integer such that 4*2^(x+2.5) [#permalink] New post   06 Jan 2016, 05:30
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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
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timothyhenman1
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Re: If x is a positive integer such that 4*2^(x+2.5) [#permalink] New post   06 Jan 2016, 11:36
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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Re: If x is a positive integer such that 4*2^(x+2.5) [#permalink] New post   06 Jan 2016, 23:41
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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.
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jojoy33
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If x is a positive integer such that 4*2^(x+2.5) [#permalink] New post   02 Dec 2016, 01:54
Question says x is a positive integer. the only possible choice where x^3 is an integer is e
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Re: If x is a positive integer such that 4*2^(x+2.5) [#permalink] New post   14 Nov 2019, 01:23
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] New post   18 Nov 2019, 09:39
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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.

Answer: E
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Re: If x is a positive integer such that 4*2^(x+2.5) [#permalink]
18 Nov 2019, 09:39
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