QuestionCalculate and simplify: $(1+\sqrt{-49})-(29 i-34)$ in the form $a+bi$ .

Studdy Solution

STEP 1

Assumptions1. We are dealing with complex numbers, where $i$ is the imaginary unit with the property $i^ = -1$ . . The given expression is $(1+\sqrt{-49})-(29 i-34)$ .
3. We need to simplify the expression and express the result in the form $a + bi$ .

STEP 2

First, we need to simplify the square root term $\sqrt{-49}$ .
The square root of a negative number can be expressed as a multiple of $i$ . In general, $\sqrt{-x} = i\sqrt{x}$ .
So, $\sqrt{-49} = i\sqrt{49}$ .

STEP 3

Now, calculate the square root of49.
$\sqrt{49} =7$ So, $\sqrt{-49} = i\sqrt{49} =7i$ .

STEP 4

Substitute the value of $\sqrt{-49}$ back into the original expression.
$(1+\sqrt{-49})-(29 i-34) = (1+7i)-(29i-34)$

STEP 5

Next, distribute the negative sign to both terms in the parenthesis $(29i-34)$ .
$(1+7i)-(29i-34) =1 +7i -29i +34$

STEP 6

Now, group the real and imaginary terms together.
$1 +i -29i +34 = (1 +34) + (i -29i)$

STEP 7

implify the real and imaginary terms.
$(1 +34) + (7i -29i) =35 -22i$ So, the simplified form of the given expression in $a + bi$ form is $35 -22i$ .

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