# Lesson: Negative Exponents

## Comment on Negative Exponents

### Hi Brent

Hi Brent
If x is an integer, is x|x|<2^x?
(1)x<0
(2)x=-10

Pls explain i couldn't understand how to substitute values for |x| should i take -ve or + ve ### In both statements you are

In both statements you are told that x is negative. So, if you're going to plug in (test) values of x, you must use negative values.

The key thing to recognize here is that 2^x IS POSITIVE for ALL values of x (positive or negative)

For example, if x = 5, then 2^x = 2^5 = 32
And if x = -3, then 2^x = 2^(-3) = 1/8

(1) If x < 0, then x is NEGATIVE
So, we get: (NEGATIVE)(|NEGATIVE|) < 2^NEGATIVE
Evaluate to get: (NEGATIVE)(POSITIVE) < POSITIVE
Simplify: NEGATIVE < POSITIVE
Perfect!

2) Same process here as for statement 1.

Here's a thread discussing this question at greater length: https://gmatclub.com/forum/if-x-is-an-integer-is-x-x-2-x-144342.html

### Hi Brent. Can you walk

Hi Brent. Can you walk through the steps with your answer to this question:

https://gmatclub.com/forum/if-5n-4n-x-0-788-then-x-n-259657.html

And why/how can you flip each fraction?

Given: 5n/(4n - x) = (0.788)^(-1)
Rewrite as: 5n/(4n - x) = 1/0.788
Flip each fraction: (4n-x)/5n = 0.788
Apply above property: 4n/5n - x/5n = 0.788
Simplify: 4/5 - x/5n = 0.788
Simplify: 0.8 - x/5n = 0.788
So: x/5n = 0.012
Multiply both sides by 5 to get: x/n = 0.06
Rewrite as fraction: x/n = 6/100 = 3/50 The main property here is as follows:
If wxyz ≠ 0 and w/x = y/z, then it must also be true that: x/w = z/y

For example, since 4/8 = 1/2, it must also be true that 8/4 = 2/1
Likewise, since 9/6 = 3/2, it must also be true that 6/9 = 2/3

So, when I got to this point: 5n/(4n - x) = 1/0.788...
...I could see that flipping both sides was going to make it much easier to solve the equation.

Does that help?

Cheers,
Brent