In which step of the expression am I simplifying it wrong?

Question

Consider this expression $$\sqrt{1-x^2}$$ We could factor the negative one out like this $$\sqrt{-1(x^2-1)}$$ Now we could use take the two factors and separate them $$\sqrt{-1}\sqrt{(x^2-1)}$$

But the root of negative one is not real, I feel really dumb asking this and I don't know anyone who can help me with this at the current moment. Thank you.

Answer 1

You are actually making two mistakes without knowing.

If the expression $\sqrt{1-x^2}$ is defined, then certainly $1-x^2\ge0$. Now when you write

$\sqrt{1-x^2}=\sqrt{(-1)(x^2-1)}$, you have the product of two negatives. And of course, factoring as a product of two square roots is not allowed.

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If you want to switch to the complex numbers, then you could write

$$\sqrt{-1}\sqrt{x^2-1}=i\sqrt{x^2-1},$$ where the square root is that of a negative.

Then

$$i\sqrt{x^2-1}=i\cdot i\sqrt{1-x^2}=-\sqrt{1-x^2}\ !$$

The apparent paradox is due to the fact that the rule "the square root of a product is the product of the square roots" doesn't hold in the complex.

Answer 2

The property :

$$\sqrt{ab} = \sqrt a \sqrt b$$

is valid, only when $a,b\geq 0$.

Thus, you cannot handle the expression $\sqrt{1-x^2}$ this way.

Take also in account that this expression is valid in $\mathbb R$ only when $1-x^2 \geq 0$.