Use the properties of complex numbers to simplify (√-49+5i)+(8−√-4).

Question

explain how you get 8+10i as your answer.

Answer 1

To simplify the expression (√-49+5i)+(8−√-4), we can start by simplifying each square root term.

√-49 = √(49i) = √(7^2 * i) = 7√i
√-4 = √(4i) = √(2^2 * i) = 2√i

Now, let's substitute these simplified values back into the original expression:

(7√i + 5i) + (8 - 2√i)

Next, let's combine like terms. We have two terms with √i and two terms with i:

(7√i - 2√i) + (5i + 8)

Simplifying further, we can combine the √i terms and the i terms:

5√i + (8 + 5i)

Now, let's convert the remaining square root term into standard form by multiplying the coefficient (5) by the square root value (√i):

5√i = 5 * (√i) = 5 * (√(1 * i)) = 5 * (√1 * √i) = 5 * (1 * √i) = 5√i

Finally, combining the terms:

5√i + (8 + 5i) = 8 + 5√i + 5i

Since the imaginary unit i is defined as √(-1), we can substitute i = √(-1):

8 + 5√i + 5i = 8 + 5√(-1) + 5√(-1)

Since √(-1) is commonly represented as i, we can simplify further:

8 + 5i + 5i = 8 + 10i

Therefore, the simplified form of (√-49+5i)+(8−√-4) is 8 + 10i.