Math  /  Algebra

QuestionSelect all of the following statements that are true A. 4i2=16i4 i^{2}=16 i B. xI(4i)2=16\mathbb{x} \mathbb{I}(4 i)^{2}=16 c. 4i=164 i=\sqrt{-16} D. \mid TVI 4=i164=i \sqrt{-16} E. 4=i16-4=i \sqrt{-16}

Studdy Solution

STEP 1

1. i i is the imaginary unit, defined as i=1 i = \sqrt{-1} .
2. i2=1 i^2 = -1 because (1)2=1 (\sqrt{-1})^2 = -1 .
3. We will evaluate each statement to determine its truth value.

STEP 2

1. Evaluate statement A.
2. Evaluate statement B.
3. Evaluate statement C.
4. Evaluate statement D.
5. Evaluate statement E.

STEP 3

Evaluate statement A: 4i2=16i 4i^2 = 16i .
First, calculate 4i2 4i^2 :
4i2=4(1)=4 4i^2 = 4(-1) = -4
Compare with 16i 16i :
Since 416i-4 \neq 16i, statement A is false.

STEP 4

Evaluate statement B: xI(4i)2=16 \mathbb{x} \mathbb{I} (4i)^2 = 16 .
First, calculate (4i)2 (4i)^2 :
(4i)2=16i2=16(1)=16 (4i)^2 = 16i^2 = 16(-1) = -16
Compare with 16 16 :
Since 1616-16 \neq 16, statement B is false.

STEP 5

Evaluate statement C: 4i=16 4i = \sqrt{-16} .
Calculate 16 \sqrt{-16} :
16=161=4i \sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i
Since 4i=4i 4i = 4i , statement C is true.

STEP 6

Evaluate statement D: TVI4=i16 \mid \text{TVI} \, 4 = i \sqrt{-16} .
Calculate i16 i \sqrt{-16} :
i16=i4i=4i2=4(1)=4 i \sqrt{-16} = i \cdot 4i = 4i^2 = 4(-1) = -4
Since 44 4 \neq -4 , statement D is false.

STEP 7

Evaluate statement E: 4=i16 -4 = i \sqrt{-16} .
We already calculated i16 i \sqrt{-16} in Step 4:
i16=4 i \sqrt{-16} = -4
Since 4=4 -4 = -4 , statement E is true.
The true statements are C and E \boxed{\text{C and E}} .

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