Question: Variables in Base and Exponent

Comment on Variables in Base and Exponent

How do we identify if we have to consider an exception for base = 1 and -1 or not??
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?
greenlight-admin's picture

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 we're only given that x≠0, are you saying it's safe to conclude that x is also not 1 and -1?

By the way, thank you for all your prompt responses thus far.
greenlight-admin's picture

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.


thanks admin, this question is also mine, but now it is clear!

I had the same questions as the other two commenters, but now I am wondering how a question involving so many stages of calculation could possibly be completed in the required 90 seconds.
greenlight-admin's picture

I think it's time-consuming, but possible to complete in that time frame.

I got the right answers, but did it in a slightly different way. Just wanted to double check if it's also another correct method to solve it.

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).
greenlight-admin's picture

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 as solution
which is not equal to 1
Hence answers should be 3,-3 and 1
Please help me understand how (-1) is a solution?
greenlight-admin's picture

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?


is -3^0=1 ? please i want to know
greenlight-admin's picture

Any non-zero number to the power of 0 equals 1.
So, (-3)^0 = 1, (-4.55)^0 = 1, (19)^0 = 1, etc


Hey Brent! How come at the beginning when you rewrote the equation, you took:


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?
greenlight-admin's picture

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

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:

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