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If 4^4x = 1600, what is the value of (4^x–1)^2?

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Yash12345
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If 4^4x = 1600, what is the value of (4^x–1)^2? [#permalink] New post   18 Oct 2013, 10:52
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If \(4^{4x} = 1600\), what is the value of \(4^{(x–1)^2}\)?

A. 40
B. 20
C. 10
D. 5/2
E. 5/4
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Re: If 4^4x = 1600, what is the value of (4^x–1)^2? [#permalink] New post   02 Nov 2013, 23:29
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ronr34 wrote:
Bunuel wrote:
Yash12345 wrote:
If 4^(4x) = 1600, what is the value of [4^(x–1)]^2?

A. 40
B. 20
C. 10
D. 5/2
E. 5/4


\(4^{4x} = 1600\) --> \(4^{2x} = 40\) -

\(4^{(x-1)^2}=4^{2(x-1)}=4^{2x-2}=\frac{4^{2x}}{4^2}=\frac{40}{16}=\frac{5}{2}\).

Answer: D.

Is there another aproach?What if I don't realize the \(4^{4x} = 1600\) --> \(4^{2x} = 40\) - ?



I think what Bunuel did is the easiest approach. However, if you are worried that this might not strike you, start with the unknown entity.

\(4^{(x-1)^2} = [\frac{4^x}{4^1}]^2 = [\frac{4^{2x}}{4^2}]\) and let \(t = [\frac{4^{2x}}{4^2}]\)

Now, given that \(4^{4x} = 1600.\) Thus,\(t^2 = [\frac{4^{4x}}{4^4}]\) =\([\frac{1600}{16*16}] = [\frac{100}{16}]\)and \(t = \frac{10}{4} =\frac{5}{2}\)
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Re: If 4^4x = 1600, what is the value of (4^x–1)^2? [#permalink] New post   18 Oct 2013, 10:59
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Yash12345 wrote:
If 4^(4x) = 1600, what is the value of [4^(x–1)]^2?

A. 40
B. 20
C. 10
D. 5/2
E. 5/4


\(4^{4x} = 1600\) --> \(4^{2x} = 40\) -

\((4^{(x-1)})^2=4^{2(x-1)}=4^{2x-2}=\frac{4^{2x}}{4^2}=\frac{40}{16}=\frac{5}{2}\).

Answer: D.
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Re: If 4^4x = 1600, what is the value of (4^x–1)^2? [#permalink] New post   Updated on: 02 Sep 2014, 21:01
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One more approach:

Refer Step I to Step VI
Attachment:
power.jpg
power.jpg [ 47.28 KiB | Viewed 16846 times ]

\(4^{4x} = 1600\)

Dividing both sides by \(4^4\)
\(\frac{4^{4x}}{4^4} = \frac{1600}{4^4}\)

\(4^{4x-4} = \frac{100}{16}\)

\(4^{(x-1)^4} = \frac{10^2}{4^2}\)

Square root both sides

\(4^{(x-1)^2} = \frac{10}{4} = \frac{5}{2}\)

Answer = D

Originally posted by PareshGmat on 24 Feb 2014, 01:28.
Last edited by PareshGmat on 02 Sep 2014, 21:01, edited 1 time in total.
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Re: If 4^4x = 1600, what is the value of (4^x–1)^2? [#permalink] New post   21 Feb 2014, 00:18
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streamingline wrote:
Bunuel wrote:
Yash12345 wrote:
If 4^(4x) = 1600, what is the value of [4^(x–1)]^2?

A. 40
B. 20
C. 10
D. 5/2
E. 5/4


\(4^{4x} = 1600\) --> \(4^{2x} = 40\) -

\(4^{(x-1)^2}=4^{2(x-1)}=4^{2x-2}=\frac{4^{2x}}{4^2}=\frac{40}{16}=\frac{5}{2}\).

Answer: D.



Hi Bunuel, why isn't (x-1)^2 is not treated like (a-b)^2 formula?

Thanks


Actually parenthesis were missing there. Edited, it should read: \((4^{(x-1)})^2=4^{2(x-1)}=4^{2x-2}=\frac{4^{2x}}{4^2}=\frac{40}{16}=\frac{5}{2}\).

If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus:
\(a^m^n=a^{(m^n)}\) and not \((a^m)^n\), which on the other hand equals to \(a^{mn}\).

So:
\((a^m)^n=a^{mn}\);

\(a^m^n=a^{(m^n)}\) and not \((a^m)^n\).

Hope it helps.
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Re: If 4^4x = 1600, what is the value of (4^x–1)^2? [#permalink] New post   27 Feb 2014, 11:08
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hieracity wrote:
Bunuel wrote:
Yash12345 wrote:
If 4^(4x) = 1600, what is the value of [4^(x–1)]^2?

A. 40
B. 20
C. 10
D. 5/2
E. 5/4


\(4^{4x} = 1600\) --> \(4^{2x} = 40\) -

\((4^{(x-1)})^2=4^{2(x-1)}=4^{2x-2}=\frac{4^{2x}}{4^2}=\frac{40}{16}=\frac{5}{2}\).

Answer: D.


Hi Bruno,

thank you for posting all these answers. They are a great tool!!

Quick question though. I just want to confirm the steps of \(4^{4x} = 1600\) TO \(4^{2x} = 40\) -

Do you just squareroot the two sides? \(\sqrt{4^{4x}} = \sqrt{1600}\)
So the base, 4, doesn't change, only the ^4x gets rooted to ^2x. Is that right?

Thank you again!


Yes:
\(4^{4x} = 1600\);

\((4^{2x})^2 = 40^2\);

\(4^{2x} =40\).

Hope it's clear.
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Re: If 4^4x = 1600, what is the value of (4^x–1)^2? [#permalink] New post   05 Feb 2019, 00:06
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KanishkM wrote:
Bunuel wrote:
Yash12345 wrote:
If 4^(4x) = 1600, what is the value of [4^(x–1)]^2?

A. 40
B. 20
C. 10
D. 5/2
E. 5/4


\(4^{4x} = 1600\) --> \(4^{2x} = 40\) -

\((4^{(x-1)})^2=4^{2(x-1)}=4^{2x-2}=\frac{4^{2x}}{4^2}=\frac{40}{16}=\frac{5}{2}\).

Answer: D.


Hi Bunuel

What is wrong in this approach?? Can you please share your thoughts on this.

If 4^(4x) = 1600

\(2^{8x} = 2^6 * 5^2\)
8x = 6
x= 3/4

Now [4^(x–1)]^2

[2^2(x–1)]^2

[ 2 ^(2x -2)] ^ 2 (substituted the value of x= 3/4 here)

[2^-1/2]^2

1/2

Here i was able to guess D, because the denominator was 2 and somehow that 5 has come to the numerator as it is the root of 25.


The red part is not correct.
If you could get 8x = 6 from here \(2^{8x} = 2^6 * 5^2\), then you'd get 1=5^2, which is obviously not correct. The point is that you cannot equate the powers there. Some power of 2 (2^(8x)) to be equal to 2^6 * 5^2, x must be some irrational number.
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Re: If 4^4x = 1600, what is the value of (4^x–1)^2? [#permalink] New post   20 Oct 2013, 06:06
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chitrasekar2k5 wrote:
Bunuel wrote:
Yash12345 wrote:
If 4^(4x) = 1600, what is the value of [4^(x–1)]^2?

A. 40
B. 20
C. 10
D. 5/2
E. 5/4


\(4^{4x} = 1600\) --> \(4^{2x} = 40\) -

\(4^{(x-1)^2}=4^{2(x-1)}=4^{2x-2}=\frac{4^{2x}}{4^2}=\frac{40}{16}=\frac{5}{2}\).

Answer: D.


I am getting the answer as 5/4.. I just can figure out what have i missed?

4^4 . 4^x = 4^3 . 5^2
4^2x = 25

now, 4^2x/4^2 = 25/16 =5/4.


\(4^{4}*4^x=4^{4+x}\) not \(4^{4x}\): \(a^n*a^m=a^{n+m}\)

Check here for more: math-number-theory-88376.html
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Re: If 4^4x = 1600, what is the value of (4^x–1)^2? [#permalink] New post   20 Oct 2013, 06:00
Bunuel wrote:
Yash12345 wrote:
If 4^(4x) = 1600, what is the value of [4^(x–1)]^2?

A. 40
B. 20
C. 10
D. 5/2
E. 5/4


\(4^{4x} = 1600\) --> \(4^{2x} = 40\) -

\(4^{(x-1)^2}=4^{2(x-1)}=4^{2x-2}=\frac{4^{2x}}{4^2}=\frac{40}{16}=\frac{5}{2}\).

Answer: D.


I am getting the answer as 5/4.. I just can figure out what have i missed?

4^4 . 4^x = 4^3 . 5^2
4^2x = 25

now, 4^2x/4^2 = 25/16 =5/4.
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Re: If 4^4x = 1600, what is the value of (4^x–1)^2? [#permalink] New post   02 Nov 2013, 03:35
Bunuel wrote:
Yash12345 wrote:
If 4^(4x) = 1600, what is the value of [4^(x–1)]^2?

A. 40
B. 20
C. 10
D. 5/2
E. 5/4


\(4^{4x} = 1600\) --> \(4^{2x} = 40\) -

\(4^{(x-1)^2}=4^{2(x-1)}=4^{2x-2}=\frac{4^{2x}}{4^2}=\frac{40}{16}=\frac{5}{2}\).

Answer: D.

Is there another aproach?What if I don't realize the \(4^{4x} = 1600\) --> \(4^{2x} = 40\) - ?
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Re: If 4^4x = 1600, what is the value of (4^x–1)^2? [#permalink] New post   03 Nov 2013, 01:30
Great!!!
Thanks a lot.
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Re: If 4^4x = 1600, what is the value of (4^x–1)^2? [#permalink] New post   20 Feb 2014, 16:43
Bunuel wrote:
Yash12345 wrote:
If 4^(4x) = 1600, what is the value of [4^(x–1)]^2?

A. 40
B. 20
C. 10
D. 5/2
E. 5/4


\(4^{4x} = 1600\) --> \(4^{2x} = 40\) -

\(4^{(x-1)^2}=4^{2(x-1)}=4^{2x-2}=\frac{4^{2x}}{4^2}=\frac{40}{16}=\frac{5}{2}\).

Answer: D.



Hi Bunuel, why isn't (x-1)^2 is not treated like (a-b)^2 formula?

Thanks
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Re: If 4^4x = 1600, what is the value of (4^x–1)^2? [#permalink] New post   27 Feb 2014, 09:32
Bunuel wrote:
Yash12345 wrote:
If 4^(4x) = 1600, what is the value of [4^(x–1)]^2?

A. 40
B. 20
C. 10
D. 5/2
E. 5/4


\(4^{4x} = 1600\) --> \(4^{2x} = 40\) -

\((4^{(x-1)})^2=4^{2(x-1)}=4^{2x-2}=\frac{4^{2x}}{4^2}=\frac{40}{16}=\frac{5}{2}\).

Answer: D.


Hi Bruno,

thank you for posting all these answers. They are a great tool!!

Quick question though. I just want to confirm the steps of \(4^{4x} = 1600\) TO \(4^{2x} = 40\) -

Do you just squareroot the two sides? \(\sqrt{4^{4x}} = \sqrt{1600}\)
So the base, 4, doesn't change, only the ^4x gets rooted to ^2x. Is that right?

Thank you again!
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Re: If 4^4x = 1600, what is the value of (4^x–1)^2? [#permalink] New post   04 Feb 2019, 23:21
Bunuel wrote:
Yash12345 wrote:
If 4^(4x) = 1600, what is the value of [4^(x–1)]^2?

A. 40
B. 20
C. 10
D. 5/2
E. 5/4


\(4^{4x} = 1600\) --> \(4^{2x} = 40\) -

\((4^{(x-1)})^2=4^{2(x-1)}=4^{2x-2}=\frac{4^{2x}}{4^2}=\frac{40}{16}=\frac{5}{2}\).

Answer: D.


Hi Bunuel

What is wrong in this approach?? Can you please share your thoughts on this.

If 4^(4x) = 1600

\(2^{8x} = 2^6 * 5^2\)
8x = 6
x= 3/4

Now [4^(x–1)]^2

[2^2(x–1)]^2

[ 2 ^(2x -2)] ^ 2 (substituted the value of x= 3/4 here)

[2^-1/2]^2

1/2

Here i was able to guess D, because the denominator was 2 and somehow that 5 has come to the numerator as it is the root of 25.
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Re: If 4^4x = 1600, what is the value of (4^x–1)^2? [#permalink] New post   05 Feb 2019, 00:14
Bunuel wrote:
KanishkM wrote:
Hi Bunuel

What is wrong in this approach?? Can you please share your thoughts on this.

If 4^(4x) = 1600

\(2^{8x} = 2^6 * 5^2\)
8x = 6
x= 3/4
.


The red part is not correct.
If you could get 8x = 6 from here \(2^{8x} = 2^6 * 5^2\), then you'd get 1=5^2, which is obviously not correct. The point is that you cannot equate the powers there. Some power of 2 (2^(8x)) to be equal to 2^6 * 5^2, x must be some irrational number.


Thank you Bunuel, I think i got your point

So basically if i had to get a value of x, i need to find a value which will completely consume RHS

Which now i realize, will be a cumbersome task.
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Re: If 4^4x = 1600, what is the value of (4^x–1)^2? [#permalink] New post   03 Feb 2022, 05:53
4^4x = 1600
(4^x)^4 = 1600
4^x = 2√10 (Dividing both sides by 4)

4^x/4 = 2√10/4 => 4^x-1 = √10/2

∴ (4^x-1)^2 = (√10/2)^2 = 10/4 = 5/2 (D)
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Re: If 4^4x = 1600, what is the value of (4^x1)^2? [#permalink] New post   23 Sep 2022, 00:53
Struggled with the question thinking that we are required to find the value of 4^[(x-1)^2]. :lol: :lol: ..
The question stem is not asking about [(4)^(x-1)]^2 :?
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Re: If 4^4x = 1600, what is the value of (4^x1)^2? [#permalink] New post   23 Sep 2022, 04:46
Yash12345 wrote:
If \(4^{4x} = 1600\), what is the value of \(4^{(x–1)^2}\)?

A. 40
B. 20
C. 10
D. 5/2
E. 5/4



If you were forced to guess, A-C can be eliminated if you realize that 4 to a power is 2 to a power and that there's no integer power of 2 that yields 1600

Posted from my mobile device
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Re: If 4^4x = 1600, what is the value of (4^x1)^2? [#permalink] New post   21 Oct 2023, 06:54
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Re: If 4^4x = 1600, what is the value of (4^x1)^2? [#permalink]
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