Issue understanding an Exponent

O

omnimath

New member
Joined
Feb 12, 2020
Messages
4
Alright, I've beaten the dead horse on this problem and I'm seeking some help.

the problem is exactly as follows

(-5x)^2 (5x)^2
---------------- Divided
(10x)^2 (-2x)^2

The solution is 25/16

I need to understand the following rule pertaining to this problem.

(-a)^n = a^n if n is even. I know that like -3^2 is positive and I realize that -3^16 is positive; because, n is positive; but, I'm lost understanding how that rule applies directly to this problem.

To take it 1 step further

HOW does (-5x)^2 = (5x)^2 ? that is what this problem is telling me on Wolfram and all of those sites. Then i am supposed to get 5x^4 from (5x)^2 and (5x)^2 ? because that - sign just "magically" disappears according to that rule? and the result is literally 5x^4 of the top piece of this problem? I'm just not seeing it please help me?
 
omnimath said:
Alright, I've beaten the dead horse on this problem and I'm seeking some help.

the problem is exactly as follows

(-5x)^2 (5x)^2
---------------- Divided
(10x)^2 (-2x)^2

The solution is 25/16

I need to understand the following rule pertaining to this problem.

(-a)^n = a^n if n is even. I know that like -3^2 is positive and I realize that -3^16 is positive; because, n is positive; but, I'm lost understanding how that rule applies directly to this problem.

To take it 1 step further

HOW does (-5x)^2 = (5x)^2 ? that is what this problem is telling me on Wolfram and all of those sites. Then i am supposed to get 5x^4 from (5x)^2 and (5x)^2 ? because that - sign just "magically" disappears according to that rule? and the result is literally 5x^4 of the top piece of this problem? I'm just not seeing it please help me?
Click to expand...

(-5x)^2 = (-5x)*(-5x) = (-5) * (x) * (-5) * (x) = [(-5) * (-5)] * (x) * (x) = (25) * (x) * (x) = (25) * (x)^2 = (5)^2 * (x) ^2 = (5x)^2
 
How about (-5)2? Do you see that (-5)*(-5) = 25 = 52?
Same idea applies to (-5x)2.

(-5x)2 (5x)2 = ?
What exponent related rules can be used to simplify this expression?
 
omnimath said:
I know that like -3^2 is positive and I realize that -3^16 is positive;
Click to expand...
Order of operations. Exponents first.
[math]-3^2 = - (3^2) = -9[/math]
[math]-3^{16} = -(3^{16}) = -43046721[/math]
Both negative.

[math](-3)^2 = (-3)(-3) = 9[/math]
[math](-3)^{16} = 43046721[/math]
Both positive.

-Dan
 
omnimath said:
HOW does (-5x)^2 = (5x)^2 ? that is what this problem is telling me on Wolfram and all of those sites. Then i am supposed to get 5x^4 from (5x)^2 and (5x)^2 ? because that - sign just "magically" disappears according to that rule? and the result is literally 5x^4 of the top piece of this problem? I'm just not seeing it please help me?
Click to expand...
(-5x)^2 = ((-1)(5x))^2 = (-1)^2 (5x)^2 = 1 (5x)^2 = (5x)^2

That's the magic.

Then (5x)^2 (5x)^2 = (5x)^(2+2) = (5x)^4. This is not the same as 5x^4; be careful with parentheses.
 
Dr.Peterson said:
(-5x)^2 = ((-1)(5x))^2 = (-1)^2 (5x)^2 = 1 (5x)^2 = (5x)^2

That's the magic.

Then (5x)^2 (5x)^2 = (5x)^(2+2) = (5x)^4. This is not the same as 5x^4; be careful with parentheses.
Click to expand...
This helped explain it thoroughly Thanks a million. I guess i kept trying to omit the -1 from existence. When really it can be isolated and squared by itself which explains how the rule works.
 
Subhotosh Khan said:
(-5x)^2 = (-5x)*(-5x) = (-5) * (x) * (-5) * (x) = [(-5) * (-5)] * (x) * (x) = (25) * (x) * (x) = (25) * (x)^2 = (5)^2 * (x) ^2 = (5x)^2
Click to expand...
This breakdown also granted me some excellent insight as to how it works. I appericiate you!
 
lev888 said:
How about (-5)2? Do you see that (-5)*(-5) = 25 = 52?
Same idea applies to (-5x)2.

(-5x)2 (5x)2 = ?
What exponent related rules can be used to simplify this expression?
Click to expand...
are you referring to the product rule?
 
omnimath said:
are you referring to the product rule?
Click to expand...
No.

He is referring to this theorem

[MATH](ab)^n = a^nb^n.[/MATH]
[MATH](-\ 5)^2 * 5^2 = \{(-\ 1) * 5\}^2 * 5^2 = \{(-\ 1)^2 * 5^2\} * 5^2 = (1 * 25)25 = 25^2 = 625.[/MATH]
 
omnimath said:
are you referring to the product rule?
Click to expand...
Hi omnimath. As Jeff just posted, lev was probably thinking of the 'Power of a product' rule. Yet, there are so many different ways to approach the exercise, it's possible also to use the 'Product' rule (along with others).

\[\frac{(-5x)^{2}\;(5x)^{2}}{(10x)^2\;(-2x)^2}\]

Those negative signs represent factors of -1. That is, -5 is the same as (-1)(5). Because both factors of -1 in the expression are squared, we may ignore them.

\[\frac{(5x)^{2}\;(5x)^{2}}{(10x)^2\;(2x)^2}\]

Note that 10 factors as (5)(2), and that leads to a common factor of 52 on top and bottom (which will cancel).

\[\frac{(5x)^{2}\;(5x)^{2}}{(5·2x)^2\;(2x)^2}\]

Apply the 'Power of a product' rule.

\[\frac{5^2 · x^2 · 5^2 · x^2}{5^2 · 2^2 · x^2 · 2^2 · x^2}\]

Cancel common factors

\[\frac{5^2}{2^2 · 2^2}\]

?
 
omnimath said:
Alright, I've beaten the dead horse on this problem and I'm seeking some help.

the problem is exactly as follows

(-5x)^2 (5x)^2
---------------- Divided
(10x)^2 (-2x)^2

The solution is 25/16

I need to understand the following rule pertaining to this problem.

(-a)^n = a^n if n is even. I know that like -3^2 is positive and I realize that -3^16 is positive; because, n is positive; but, I'm lost understanding how that rule applies directly to this problem.

To take it 1 step further

HOW does (-5x)^2 = (5x)^2 ? that is what this problem is telling me on Wolfram and all of those sites. Then i am supposed to get 5x^4 from (5x)^2 and (5x)^2 ? because that - sign just "magically" disappears according to that rule? and the result is literally 5x^4 of the top piece of this problem? I'm just not seeing it please help me?
Click to expand...
As pka already pointed out, -3^2 is NOT 9. -3^16 is negative. I just felt that you needed to see this again.

I suspect that you would think that 10-3^2 =1. That is because -3^2 = -9 not 9!
 
You must log in or register to reply here.
Top