Simplify The Following Expression: $\[\sqrt{-16}+\sqrt{-25}+5\\]A. \[$5 + 9i\$\] B. \[$14 + 0i\$\] C. \[$5 + \sqrt{41i}\$\] D. \[$9 - 5i\$\]

**Simplify the Expression: A Step-by-Step Guide** =====================================================

Understanding the Problem

The given expression involves the sum of two square roots and a constant. To simplify this expression, we need to understand the properties of square roots and complex numbers.

What are Square Roots?

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 × 4 = 16.

What are Complex Numbers?

Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation i^2 = -1.

Simplifying the Expression

To simplify the given expression, we need to evaluate the square roots of -16 and -25.

Evaluating the Square Roots

The square root of -16 can be written as:

√(-16) = √(-1 × 16) = √(-1) × √16 = i × 4 = 4i

The square root of -25 can be written as:

√(-25) = √(-1 × 25) = √(-1) × √25 = i × 5 = 5i

Substituting the Values

Now, we can substitute the values of the square roots into the original expression:

√(-16) + √(-25) + 5 = 4i + 5i + 5

Combining Like Terms

We can combine the like terms (the imaginary terms) by adding their coefficients:

4i + 5i = 9i

Final Answer

Therefore, the simplified expression is:

9i + 5

Answer Options

Now, let's compare our answer with the given options:

A. 5 + 9i B. 14 + 0i C. 5 + √41i D. 9 - 5i

Our answer, 9i + 5, is equivalent to option A, 5 + 9i.

Q&A

Q: What is the square root of -16? A: The square root of -16 is 4i.

Q: What is the square root of -25? A: The square root of -25 is 5i.

Q: How do you simplify the expression √(-16) + √(-25) + 5? A: To simplify the expression, we need to evaluate the square roots of -16 and -25, and then substitute the values into the original expression.

Q: What is the final answer? A: The final answer is 9i + 5, which is equivalent to option A, 5 + 9i.

Conclusion

In this article, we simplified the expression √(-16) + √(-25) + 5 by evaluating the square roots of -16 and -25, and then substituting the values into the original expression. We also compared our answer with the given options and found that it is equivalent to option A, 5 + 9i.