To simplify the expression [tex]\(\sqrt{-25} - 4\)[/tex], we need to address the square root of a negative number, which involves the concept of imaginary numbers. Here is the step-by-step process:
1. Identify the imaginary component:
- The square root of a negative number can be dissected into the product of the square root of the positive counterpart and the square root of -1, denoted by the imaginary unit [tex]\(i\)[/tex].
2. Simplify [tex]\(\sqrt{-25}\)[/tex]:
- Write [tex]\(-25\)[/tex] as [tex]\(25 \times (-1)\)[/tex].
- Now, take the square root of each part separately:
[tex]\[
\sqrt{-25} = \sqrt{25 \times (-1)} = \sqrt{25} \times \sqrt{-1}
\][/tex]
- The square root of 25 is 5, and the square root of -1 is the imaginary unit [tex]\(i\)[/tex]:
[tex]\[
\sqrt{25} = 5, \quad \sqrt{-1} = i
\][/tex]
- Therefore, we get:
[tex]\[
\sqrt{-25} = 5i
\][/tex]
3. Subtract 4 from the result:
- Now, the original expression can be rewritten as:
[tex]\[
5i - 4
\][/tex]
4. Write the expression in standard form:
- The standard form for a complex number is [tex]\(a + bi\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are real numbers. In this case, [tex]\(a = -4\)[/tex] and [tex]\(b = 5\)[/tex].
- Thus, the simplified form of the expression is:
[tex]\[
-4 + 5i
\][/tex]
So, the simplified form of the expression [tex]\(\sqrt{-25} - 4\)[/tex] is [tex]\(\boxed{-4 + 5i}\)[/tex].
