To solve the expression [tex]\(\frac{\sqrt{25a}}{\sqrt{-25a}}\)[/tex], let's break it down into simpler steps:
1. Simplify the numerator:
The expression in the numerator is [tex]\(\sqrt{25a}\)[/tex].
We can simplify this as:
[tex]\[
\sqrt{25a} = \sqrt{25} \cdot \sqrt{a}
\][/tex]
Since [tex]\(\sqrt{25} = 5\)[/tex], the simplified numerator is:
[tex]\[
5\sqrt{a}
\][/tex]
2. Simplify the denominator:
The expression in the denominator is [tex]\(\sqrt{-25a}\)[/tex].
We can simplify this as:
[tex]\[
\sqrt{-25a} = \sqrt{-1 \cdot 25a} = \sqrt{25a} \cdot \sqrt{-1}
\][/tex]
Again, [tex]\(\sqrt{25} = 5\)[/tex], and [tex]\(\sqrt{-1} = i\)[/tex] (where [tex]\(i\)[/tex] is the imaginary unit). Therefore, the simplified denominator is:
[tex]\[
5i\sqrt{a}
\][/tex]
3. Combine the simplified numerator and denominator:
Now we have:
[tex]\[
\frac{\sqrt{25a}}{\sqrt{-25a}} = \frac{5\sqrt{a}}{5i\sqrt{a}}
\][/tex]
Notice that [tex]\(5\sqrt{a}\)[/tex] in the numerator and denominator cancels out, so we are left with:
[tex]\[
\frac{5\sqrt{a}}{5i\sqrt{a}} = \frac{1}{i}
\][/tex]
4. Simplify [tex]\(\frac{1}{i}\)[/tex]:
To simplify [tex]\(\frac{1}{i}\)[/tex], we multiply the numerator and the denominator by the complex conjugate of the denominator, which is [tex]\(-i\)[/tex]:
[tex]\[
\frac{1}{i} \cdot \frac{-i}{-i} = \frac{-i}{-i^2}
\][/tex]
Since [tex]\(i^2 = -1\)[/tex], this becomes:
[tex]\[
\frac{-i}{(-1)} = i
\][/tex]
So, our final simplified result is:
[tex]\[
\frac{\sqrt{25a}}{\sqrt{-25a}} = i
\][/tex]
Or more generally (considering the presence of similar patterns of roots in the answer):
[tex]\[
\frac{\sqrt{a}}{\sqrt{-a}}
\][/tex]
This retains the simplified and factored form:
[tex]\[
\sqrt{a}/\sqrt{-a} = i
\][/tex]
