Question: Variables in Base and Exponent

Comment on Variables in Base and Exponent

How do we identify if we have

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?﻿

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

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.

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

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

I had the same questions as

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.

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

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).

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

Nice explanation. Thanks

I have a doubt regarding -1

I have a doubt regarding -1 as solution
-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

is -3^0=1 ? please i want to know

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

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

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