What is the square root of $(-5)\cdot(-5)$ and how is it different from $\sqrt{(-5)^2}$?

Question

What is the square root of $(-5)\cdot(-5)$ and how is it different from $\sqrt{(-5)^2}$? Can anybody explain?

Answer 1

First of all let us get to $$(-5)^2=25$$

Well $25$ has two square roots Positive one which we denote as $$\sqrt {25} =5$$ and a negative one which is denoted by $$-\sqrt {25} =-5$$ A common mistake happens when someone writes $$\sqrt {25} = \pm 5$$

Now the equation $$x^2=25$$ has two solutions and we may write them as $$x=\pm 5$$ One good formula to have in mind is that $$\sqrt {x^2} = |x|$$ which results in $$-\sqrt {x^2} = -|x|$$

Answer 2

We may differentiate "two kinds" of square rooting:

arithmetical root It belongs to questions like $\sqrt{25}$, $\sqrt{49}$, $\sqrt[3]{-8}$ with answers 5 , 7 and -2 respectively. It will always be as same sign of the argument.

algebraical root

It belongs to the set of numbers which satisfies some equation like $x^2=4$ with values $2$ and $-2$, or $x^2=25$ with values $5$ and $-5$.

Answer 3

Every positive real number has exactly two distinct square roots. They are opposites.

If $r$ is a positive real number, the symbol "$\sqrt{r}$" denotes the positive square root of $r$, and so "$-\sqrt{r}$" denotes its opposite, the root which is negative.

Note that "positive real number" does not include zero. For the special case $r=0$, there is only one square root, which is again zero. This is just because $0$ and $-0$ are the same number. By convention "the" square root is the positive root (or the only root, in the case of $0$).

However, this convention vanishes when the roots aren't real, simply because complex numbers in general don't have signs; they aren't "positive" or "negative" (unless they are actually real numbers).

Negative real numbers (and generally speaking, complex numbers which are not nonnegative real numbers) do not have real square roots. Still, it is true that every nonzero complex number $c$ has exactly two square roots, and they are opposites. In theory, they could be denoted by "$\sqrt{c}$" and "$-\sqrt{c}$"; the only problem is that there is no natural way to decide which one is represented by the symbol "$\sqrt{c}$" (once that's decided, the other root is always just its opposite).

For example, $1-i$ and $-1+i$ are the two square roots of $-2i$ because $(1-i)^2=-2i$ and $(-1+i)^2=-2i$. They are opposites, as always, but which one is $\sqrt{-2i}$? There is no universal agreement.

Finally, if you don't specify which is which, you can certainly speak of both square roots together as "$\pm\sqrt{c}$" and leave the ambiguity alone.

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Answering your question: both symbols you wrote are $5$. That's just because $(-5)\cdot(-5)$ and $(-5)^2$ are both the number $25$, and "the" square root of $25$ is $5$.

Answer 4

The fundamental theorem of algebra tells us that a polynomial of degree $n$ has $n$ roots. For your example, the polynomial is $x^2 - 25$. It is of degree 2 (because that's the highest exponent) and it therefore has two roots, two values of $x$ such that $x^2 - 25 = 0$. Those are $x = -5$ and $x = 5$.

However, we have agreed, for convention through several decades if not centuries, that the square root symbol is for a function that gives us the principal square root. In your example that would be 5, not $-5$. Maybe that feels arbitrary to you, but that's the convention.

Hence, given a real number $c$, we find that $\sqrt{c^2} = \sqrt{(-c)^2}$.