1) It's true if x and y are positive. (99% of the time when you're first learning radical simplifications, you ONLY deal with positive variables.)
2) It's true if x is positive and y is negative: but then you're dealing with imaginary numbers.
3) It's true if x is zero and y is anything other than zero. (That's fairly trivial when you thihnk about it.)
4) It's true if x is negative and y is positive. Similar to case #2....and it's dealing with imaginary numbers.
And, conversely:
1) It's false if y is zero, simply because neither side will be a real number, and the property "equals" is ONLY defined when you're comparing two real numbers. (It can be defined for complex numbers too, but you generally don't use "=" in the traditional sense when talking about 1/0.)
2) It's also false if x and y are both negative. The reason will be clearer when you use imaginary numbers, which involve the square root of -1. The square root of -1 is often symbolized as i, and it's NOT equal to -1! Why? Because -1 is one of the two square roots of +1...because -1 * -1 = +1.
And that's also why this equality won't work with two negative numbers. Usually when you say SQRT(x), it's assumed you mean the PRINCIPAL (or positive) square root. SQRT(x) will NEVER be negative.
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if all your " numbers " have to be real then it is false...x = -4 , y = - 1
if you are working with complex numbers then the answer is true
What everyone is trying to say is:
1) It's true if x and y are positive. (99% of the time when you're first learning radical simplifications, you ONLY deal with positive variables.)
2) It's true if x is positive and y is negative: but then you're dealing with imaginary numbers.
3) It's true if x is zero and y is anything other than zero. (That's fairly trivial when you thihnk about it.)
4) It's true if x is negative and y is positive. Similar to case #2....and it's dealing with imaginary numbers.
And, conversely:
1) It's false if y is zero, simply because neither side will be a real number, and the property "equals" is ONLY defined when you're comparing two real numbers. (It can be defined for complex numbers too, but you generally don't use "=" in the traditional sense when talking about 1/0.)
2) It's also false if x and y are both negative. The reason will be clearer when you use imaginary numbers, which involve the square root of -1. The square root of -1 is often symbolized as i, and it's NOT equal to -1! Why? Because -1 is one of the two square roots of +1...because -1 * -1 = +1.
And that's also why this equality won't work with two negative numbers. Usually when you say SQRT(x), it's assumed you mean the PRINCIPAL (or positive) square root. SQRT(x) will NEVER be negative.
This is true if and only if x is greater or equal to zero and y is strictly greater than zero.
be very careful about your domains.
for complex numbers, I imagine that the only restriction is that the length of y must be positive
True only when x, y ⥠0
âx / ây
= â(x / y)
It's one of those common techniques you use.
lets say you have â(16/5)
â(16/5)
= â16 / â5
= 4 / â5....so we can split it up....
Hope this helps :D
Apple-so-loot-ly true.
.. (x)^(1/2) / (y)^(1/2)
= (x)^(1/2) * (y)^(-1/2)
= (x)^(1/2) * (1/y)^(1/2)
= (x * 1/y)^(1/2)
= (x/y)^(1/2) â true for any { x, yâ 0 }, positive, negative, real or imaginary.