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For \(i=\sqrt{-1}\), which of the following complex numbers is equivalent to \(\left(10 i-4 i^2\right)-(7-3 i) ?\) A) \(-11+7 i\) B) \(-3+13 i\) C) \(3-13 i\) D) \(11-7 i\)

Short Answer

Expert verified
B) \(-3 + 13i\)

Step by step solution

01

Rewrite the expression using \(i=\sqrt{-1}\).

The provided expression is \(\left(10i - 4i^2\right) - (7 - 3i)\). Since \(i^2 = -1\), replace all instances of \(i^2\) in the given expression: \[\left(10i - 4(-1)\right) - (7-3i).\]
02

Simplify the expression

Now, simplify the expression by doing the arithmetic operations as: \[\left(10i + 4\right) - (7-3i) = 4 + 10i - 7 + 3i.\]
03

Combine the real and imaginary parts

Combine the real parts (constant terms) and the imaginary parts (terms with \(i\)): \[(4-7) + (10i+3i) = -3 + 13i.\]
04

Compare the result with the given options

We have found that the simplified complex number is \(-3 + 13i\). Comparing this to the given options, we can see that this matches option B. Therefore, the correct answer is: B) \(-3 + 13i\).

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