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25 Nov 2021, 03:45
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D
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Difficulty:
55%
(hard)
Question Stats:
55% (01:20) correct
45%
(01:13)
wrong
based on 501
sessions
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If x is not 0, for how many integers is \(|x|^x = x*x\)?
A. 0
B. 1
C. 2
D. 3
E. >3
A. 0
B. 1
C. 2
D. 3
E. >3
25 Nov 2021, 07:29
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Bunuel
Given: \(|x|^x = x*x\)
In other words: \(|x|^x = x^2\)
At this point it looks like \(x = 2\) will be a solution, since \(|2|^2 = 2^2\)
What about \(x = -2\)?
Let's test this one to get: \(|-2|^{-2} = (-2)^2\)
Simplify to get: \(\frac{1}{2}=4\). Doesn't work.
So, \(x = -2\) is not a solution.
Next, since \(1^k = 1\) for all values of k, we can see that \(x = 1\) will be a solution, since \(|1|^1 = 1^2\)
What about \(x = -1\)?
Let's test this one to get: \(|-1|^{-1} = (-1)^2\)
Simplify to get: \(1=1\) WORKS
So, \(x = -1\) is a solution.
At this point we have three possible values for \(x\): \(x = 2\), \(x = 1\), and \(x = -1\)
Although I'm not entirely sure I've identified all possible solutions, I'll give myself another 20 seconds to find another value....
.... no luck.
With fingers crossed, I'll select answer choice D
General Discussion
08 Jul 2023, 05:54
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Given that x is not 0 and we need to find for how many integers is \(|x|^x = x*x\)
We can solve this by Substitution
\(|x|^x = x*x\) = \(x^2\)
Definitely 1 and 2 are two possible choices
As we have Absolute value on left side and square on right side so let's try with -1 and -2 also
Now, there will be no other integer for which we will have \(|x|^x = x^2\)
As, x raised to its own power will always be greater or smaller than x to the power of 2
So, Answer will be D
Hope it helps!
Watch the following video to learn the Basics of Absolute Values
We can solve this by Substitution
\(|x|^x = x*x\) = \(x^2\)
Definitely 1 and 2 are two possible choices
| Case x = 1 or x = 2 | |
| -x = 1 => \(|1|^1 = 1^2\) => 1 = 1 => x = 1 is a SOLUTION | -x = 2 => \(|2|^2 = 2^2\) => 4 = 4 => x = 2 is a SOLUTION |
As we have Absolute value on left side and square on right side so let's try with -1 and -2 also
| Case x = -1 or x = -2 | |
| -x = -1 => \(|-1|^{-1} = (-1)^2\) => 1 = 1 => x = -1 is a SOLUTION | -x = -2 => \(|-2|^{-2} = (-2)^2\) => \(1/4\) ≠ 4 => x = -1 is NOT a SOLUTION |
Now, there will be no other integer for which we will have \(|x|^x = x^2\)
As, x raised to its own power will always be greater or smaller than x to the power of 2
So, Answer will be D
Hope it helps!
Watch the following video to learn the Basics of Absolute Values
