Radicals

[Denis]

(-1)^4... That would be "-1"

**No John; (-1)^4 = +1
However, -1^4 = -1 : same as -(1)^4) = -1

As you can see, brackets are quite important.

....then the simplication of sample #3: (-xy^2)^4 should be (-x^4y^8). pka shows it to be: (x^4y^8). This is what I question. What happened to the minus sign in pka's third sample?

**John, the (-xy^2)^4 works the same as (-1)^4, so equals +x^4y^8;
however, -(xy^2)^4 = -x^4y^8

**notice that the minus sign is INSIDE the brackets.

 

[John Whitaker]

The matter that confused me: how two terms identical in structure except that one exponent was 3, and the other exponent was 4, should result in one answer being positive, and the other answer being negative. I refer back to pka's three examples (above); specifically, the first term of the 2nd & 3rd examples.
Before pka threw in the towel, he wrote something that indicates even number exponents are treated differently than odd number exponents. I didn't understand the "n and j" parts, but...
Thanks anyway.
John Whitaker

 

[stapel]

(-1)^4... That would be "-1"

Really? (-1)(-1)(-1)(-1) = [(-1)(-1)][(-1)(-1)] = (+1)(+1) = +1 equals -1?

Eliz.

 

[John Whitaker]

Thank you Denis, Eliz, and pka. Despite my poorly constructed original question (which I have edited), your tenacity has provided me with enough to return to my book a bit wiser. Somewhere, I hope it will bring me to pka's "(-x^2)^4=(-x)^2j=[(-x)^2]^j... etc. Let's close the door on this one. Thanks again. (See you in the future. Stop shuddering!)

 

[Denis]

John, when this dawns on you, you'll slap yourself on the forehead real HARD!
Plus you'll realise why it's difficult to teach here, by typing...

If you can see the 3 samples pka gave me, look at the first parenthetical term in the second sample... and the same in the third.
RE: The second: (-xy^2)^4 If the simplication of that is -x^3y^6, then the simplication of sample #3: (-xy^2)^4 should be (-x^4y^8). pka shows it to be: (x^4y^8). This is what I question. What happened to the minus sign in pka's third sample?

The second pka example is (-xy^2)^3, not (-xy^2)^4 : typo?
And simplification is -x^3y^6, same as -(x^3y^6)
Works same as if (-xy^2) was (-1): so (-1)^3 = -1, same as -(1)
-1 * -1 * -1 = -1
-xy^2 * -xy^2 * -xy^2 = -x^3y^6

The third pka example is (-xy^2)^4
And simplification is x^4y^8, NOT -x^4y^8
Works same as if (-xy^2) was (-1): so (-1)^4 = +1
-1 * -1 * -1 * -1 = +1
-xy^2 * -xy^2 * -xy^2 * -xy^2 = +x^4y^8

You also posted:
"Before pka threw in the towel, he wrote something that indicates even number exponents are treated differently than odd number exponents. I didn't understand the "n and j" parts, but... "

If an integer is even, then it is divisible by 2;
using n as integer means the integer is odd or even: can't tell, right?
so to ensure we're working with an even integer, we let n = 2j:
regardless of what j is, 2j is even; if j=3 : 2j=6=n.
Similarly, to ensure n is odd, we let n = 2j+1:
regardless of what j is, 2j+1 is odd; if j=3 : 2j=6+1=7=n.

Your "edited original" still has:
"Without LaTex, how do I express the Radical Sign?
My problem is: I have an open parenthesis... "3x"... then a Radical Sign with "4y" as a Radicand... close parenthesis... ^2 "

Without LaTex (which I'm too lazy to use!), use sqrt() for radical sign.
So your problem would be shown this way: (3x * sqrt(4y))^2
And that's same as (3x)^2 * (sqrt(4y))^2 = 9x^2 * 4y = 36x^2y: right?

 

[John Whitaker]

After 'closing the door' I decided to check one more time and found Denis' reply. I was through with algebra for the day so I just skimmed it... printed it, and put it on my desk for my next assault. Still, snatches of the recent forum would swim in and out.
I recalled working out (-x)^4 on my own: "-x*-x*-x*-x" and getting as far as the first two terms... "-x*-x". And, I recalled that minus times minus equals plus... but I didn't carry that to the next two terms until I read Eliz's response to my erroneous answer to hers: "What is (-x)^4?"
Seeing how she bracketed the terms, "= [(-1)(-1)][(-1)(-1)] = (+1)(+1) = +1" the first seed of understanding began to bloom. Still, I didn't put it all together. PKA's "^nj" continued to irritate me... invading my thoughts whenever it pleased.
Then I decided to check the forum one last time and found: "John, when this dawns on you, you'll slap yourself on the forehead real HARD! " from Denis.
This proved prophetic. At 3:30 am I was invaded by "light" in the darkness of my bedroom. I got it. It all came together, and I couldn't get back to sleep. (That's OK... I watched a Bogart movie on TV.)
Eliz, PKA, Denis... thanks a lot. Hope to see you all next subject!!! Thanks again.
John