Last year in Pre-Algebra we learned about square roots. I was taught then that
√64=8 and √100=10, which I understood and accepted. I was also taught that ±√64=8,−8 because both of those numbers squared is 64, which I also get.
But this year, with a new school and teacher in a different state, our teacher is telling us that:
√64=8,−8 and ±√64 also is 8,−8. The way to get the positive root of something is:
And these seem to contradict each other. I was always taught that a regular square root returned a positive number and only a positive number, but now my teacher is saying a regular square root gives two numbers, and considering the square root of a number n is defined as
y2=n I see where he is coming from.
Upon researching this Wikipedia says:
For example, 4 and −4 are square roots of 16 because 42=(−4)2=16
And Wolfram MathWorld says:
Note that any positive real number has two square roots, one positive and one negative. For example, the square roots of 9 are −3 and +3
But on the other side, Wolfram Alpha, when given “The square root of 9” gives only 3.
So, which is right? Is √64 considered 8? or is it 8,−8?
Your new teacher is wrong. √⋅ is the principal square root operator. That means it returns only the principal root — the positive one. √64=8. It does NOT equal −8.
On the other hand, the equation 64=x2 DOES have 2 solutions: x=8 or x=−8. Thus both 8 and −8 are square roots of 64.
Let’s see what happens when we take the principal square root of both sides of this equation: 64=x2⟹√64=√x2⟹8=|x|⟹x=8 or x=−8
Thus the fact that the principal square root operation throws out the negative root isn’t much of a problem as the math still works out correctly.