Rational Exponents

Hockeyman

Junior Member
Joined
Dec 8, 2005
Messages
79
I just wanted to make sure a few of these problems were right.

1) (25x^-6 y^2) ^ 1/2 = SqRt [ 25x^-6 y^2 ] = 5x^-3 y = 5y/x^3

2) ( 8a^-3 b^9)^ 2/3 = CubedRt of [8^2] a^-2 b^6 = 4a^-2 b^6= 4b^6/a^2

3) ([16z^4] / [25x^8])^ -1/2 = {[16^-1/2] [z^-2]} / {[25^-1/2] [x^-4]} = [5x^4] / [4z^2]

Are these correct?
 

tkhunny

Moderator
Staff member
Joined
Apr 12, 2005
Messages
9,991
A fundamental and oft'-overlooked principle of Real Numbers is missing.

sqrt(x^2) = |x| in the absence of additional information about 'x'.
If you KNOW x >= 0, then sqrt(x^2) = x.
If you do not KNOW x >= 0, then sqrt(x^2) = |x|

These are correct:

(x^6)^(1/2) = |x^3| -- x^3 could be positive or negative.
(x^8)^(1/2) = x^4 -- x^4 cannot be negative.
sqrt(25) = 5 -- 5 > 0

This makes your #1 a little off.
#2 is easier, since cube roots don't do the same thing as even roots.
#3 is OK, but it may have been luck.
 
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