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Comment on Variables in Base and Exponent
How do we identify if we have
While solving this question, I considered 3 and -3 as the only answer since the rule says that base cannot be 1 and -1. Thinking it could be a trap answer. Could you please explain?
If we're given a VARIABLE for
If we're given a VARIABLE for the base, then we need to consider whether or not that variable is 0, 1 or -1
For example, if we're told that x^y = x^3, then we can't automatically conclude that y = 3.
We can't make this conclusion, because there are certain values of x that don't work.
For example, if x = 1, the equation becomes 1^y = 1^3, in which case y can equal ANY number.
Likewise, if x = 0, the equation becomes 0^y = 0^3, in which case y can equal ANY number.
Alternatively, if we're explicitly told the value of the base AND that value is not 0, 1 or -1, then we can make solid conclusions. For example, if we're told that 5^x = 5^7, then we can be certain that x = 7
For this type of question, if
By the way, thank you for all your prompt responses thus far.
No, we can't make that
No, we can't make that conclusion.
In fact, even though x ≠ 0, it turns out that x = 1 and x = -1 ARE possible solutions to the equation.
Cheers,
Brent
thanks admin, this question
I had the same questions as
I think it's time-consuming,
I think it's time-consuming, but possible to complete in that time frame.
I got the right answers, but
1) At first I took 1 from each side:
x^(x^2-9) - 1 = 0
2) I figured that "x^(x^2-9)" would have to equal 1 since (1-1 = 0)
3) Then I just plugged in the numbers (9 was eliminated easily since the exponent was obviously too high). This method didn't seem to take too long to do, which was why I wanted to confirm if it's a correct way (as it's a "hard" question, but seemed too easy).
That's a perfectly valid
That's a perfectly valid approach!
ASIDE: Steps 1 and 2 are really necessary. You used those steps to conclude that x^(x^2-9) = 1, even though that is the original equation :-)
Nice explanation. Thanks
I have a doubt regarding -1
-1^-8
=-[1/(1^8)]
=-[1/1]
=-1
which is not equal to 1
Hence answers should be 3,-3 and 1
Please help me understand how (-1) is a solution?
Thanks.
Be careful. (-1)^(-8) does
Be careful. (-1)^(-8) does not equal = -[1/(1^8)]
The rule for negative exponents is: b^(-n) = 1/(b^n)
So, (-1)^-8 = 1/(-1)^8
= 1/1
= 1
Aside: Let's take a closer look at (-1)^8
(-1)^8 = (-1)(-1)(-1)(-1)(-1)(-1)(-1)(-1)
Find the product in PAIRS
(-1)(-1)(-1)(-1)(-1)(-1)(-1)(-1) = (1)(1)(1)(1)
= 1
Does that help?
Cheers,
Brent
is -3^0=1 ? please i want to
Any non-zero number to the
Any non-zero number to the power of 0 equals 1.
So, (-3)^0 = 1, (-4.55)^0 = 1, (19)^0 = 1, etc
Cheers,
Brent
Hey Brent! How come at the
x^(x^2-9)=1
and changed the right side to x^0? I don't understand why you did that.
WHEW, it gets confusing writing exponents within exponents when you have to use ^.
Also, one other question, I knew that 1 raised to any power was equal to 1, but I didn't know if that same rule applied to -1. So is the rule that 1 OR -1 raised to any power is 1? If -1 was raised to an odd power, would it be -1?
The goal with these kinds of
The goal with these kinds of questions is to rewrite expressions so that they have the same bases.
For example, we can take: 5^(x-4) = 25...
...and rewrite 25 to get: 5^(x-4) = 5^2
We can then write: x - 4 = 2
Etc.
The same applies to the following equation: x^(x^2-9) = 1
Does x does not equal 0, we know that x^0 = 1
So, we rewrite our equation as: x^(x^2-9) = x^0, at which point we can conclude that x^2 - 9 = 0
That's formalize the properties regarding 1 and -1 in the base:
PROPERTY #1: 1^k = 1 for ALL values of k
PROPERTY #2: (-1)^k = 1 for all EVEN values of k
PROPERTY #3: (-1)^k = -1 for all ODD values of k
For more on these properties, watch: https://www.greenlighttestprep.com/module/gre-powers-and-roots/video/1021