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Question
2^(x+4) - x^(x+2)=3?

m^n=121 ,m and n whole nos then (m-1)^n+1=?

2^x=3^y=6^-z then 1/x+1/y+1/z=?

[16^(x+1) +20(4^2x)]/[2^(x-3)][8^(x+2)]

Answer
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2^(x+4) - x^(x+2)=3?

First note that x must be small, for large numbers don't have powers very close together.
For the first one, note that 2^2 = 4, and 3 is 1 less.
So if we had 4-1, that would be 3.
To get 2^(x+4) = 4, we would need x as -2, so that would give 2^2.
To get x^(x+2) = 1, we would need x as -2, for -2^0 = 1.
Thus, x = -2 is the answer.

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m^n=121 ,m and n whole nos then (m-1)^n+1=?

If m^n = 121, then note that 121 = 11^2, so m = 11 and n = 2.
If we are trying to find(m-1)^n + 1, that would be 10^2 + 1 = 101.
If it is really suppose to be (m-1)^(n+1), that would be 10^3 = 1000.

Also note that m=-11 works, since -11^2 = 121.
If that were the case, then the result would be -11^2 + 1 or -11^3.
That would be -1,21 + 1 =-120 or -11^3 = -1,331.

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2^x=3^y=6^-z then 1/x+1/y+1/z=?

If 2^x = 3^y = 6^-z, then take the ln() of all terms.
That gives ln(2^x) = ln(3^y) = ln(6^z).
It is known that ln(x^y) = y*ln(x), so this can be rewritten as x*ln(2) = y*ln(3) = z*ln(6).
Since 6 = 2*3, this can be rewritten as x*ln(2) = y*ln(3) = z*[ln(2)+ln(3)].
So it is known that y = x * ln(2) / ln(3) and z = x * ln(2) / [ln(2) + ln(3)].
All we need is to choose x, then y and z are given.

There are no integer solutions since some power of 3 never exist as being equal unless
x=0, y=0, and z=0, for then we would have 1 = 1 = 1.  If that were the case, then
1/x + 1/y + 1/z = 1/0 + 1/0 + 1/0 = undefined + undefined + undefined = undefined.



Now if this problem were changed to one with integer solutions ...

Now if it was x^2 = y^3 = z^6, they would have integer solutions.
They would be x = a^3, y = a^2, and z = a, for that would give
(a^3)^2 = (a^2)^3 = a^6, which is a^6 = a^6 = a^6.

If a = 1, that gives 1, 1, and 1, with each term being 1.
The answer would be undefined.

If a = 2, that gives 8, 4, and 2, with each term being 64.
The answer would be 1/8 + 1/4 + 1/2 = 7/8.

If a = 3, that gives 27, 9, and 3, with each term being 729.
The answer would be 1/27 + 1/9 + 1/3 = 13/27.

If a = 4, that gives 64, 16, and 4, with each term being 4,096.
The answer would be 1/64 + 1/16 + 1/4 = 21/64.

If a = 5, that gives 125, 25, and 5, with each term being 15,625.
The answer would be 1/125 + 1/25 + 1/5 = 31/125.


If there is something that is written incorrectly, ask again.

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16^(x+1) +20(4^2x)]/[2^(x-3)][8^(x+2)]

Note that [16^(x+1) +20(4^2x)]/[2^(x-3)][8^(x+2)] should really be written as
[16^(x+1) + 20(4^{2x})]/{[2^(x-3)][8^(x+2)]}.

I added '{' and '}' around both terms after the '/' symbol or else would do the division and then multiply all that by that last term.  This is a fact that often occurs because on the PC, the problem needs to be on one line and the last term is really suppose to be in the denominator.

I also added '{' and '}' in the 4^2x.  Since powers have precedence over multiplication,
as written we would have 16x.

I also put a ' ' after that + on top, just to clarify the problem.
The answer would be no different, but I like to put spaces around addition/subtraction terms to designate that each term is calculated individually.


For [16^(x+1) + 20(4^2x)]/{[2^(x-3)][8^(x+2)]}, it all depends on the choice of x.

If x = 0, we have (16 + 1) / [(1/8)(64)] = 17 / 8 = 2.125.

If x = 1, we have (16^2 + 20*4^2)/[(1/4)(1)] = (256 + 320)*4 = 576*4 = 2,304.
If you recognize 576 as being 24^2 and 4 as being 2^2, then note the answer would be 48^2,
for that is what 2,304 is. { I said this because 2,304 is recognized as 48^2. }.

If x = 2, we have (16^3 + 20*4^4)/(2*8^4). = (4,096 + 5,120)/8,192 = 9,216/8,192.
Note that 9,216 = 96^2, and that 96=3*2^5, so 9,216 = (3*2^5)^2 = 3^2 * 2^10.
This makes 9,216/8,192 = 9/8.

If x = 3, we have (16^4 + 20*4^6)/(1*8^6) = (65,536 + 81,920)/262,144 = 147,456/262,144.
Note that the top and bottom can be reduced by dividing them both by 2, many times.
If is 147,456/262,144 = 75,728/131,072 = 37,864/65,536 = 18,932/32,768 = 9,466/16,384 =
4,733/8,192.  Now the denominator is 2^13, and the numerator is not divisible by 2,
so that would be the final answer is x = 3.

If x = -1, it would be really simple.  It would be [1 + 20/16]/[(1/16)8] = (9/4)/(1/2) = 9/2.

If there is more to the last problem, ask again ... after Thanksgiving.  

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