QuestionSimplify: $(7-2 i)^{2}-(25-12 i)$ . Choose the correct answer: a. $28+21 i$ , b. $28-16 i$ , c. $20-40 i$ , d. $20-16 i$ .

Studdy Solution

STEP 1

Assumptions1. We are dealing with complex numbers, where $i$ is the imaginary unit with the property $i^{} = -1$ . . The operations of addition, subtraction, and multiplication apply to complex numbers in the same way as they do to real numbers.

STEP 2

First, we need to simplify the square of $(7-2i)$ . We can do this by multiplying the complex number by itself.
$(7-2i)^{2} = (7-2i) \times (7-2i)$

STEP 3

Now, let's expand the expression using the distributive property of multiplication over addition.
$(7-2i) \times (7-2i) =7 \times7 +7 \times (-2i) + (-2i) \times7 + (-2i) \times (-2i)$

STEP 4

implify the expression by performing the multiplication.
$(7-2i) \times (7-2i) =49 -14i -14i +4$

STEP 5

Combine like terms.
$(7-2i) \times (7-2i) =53 -28i$

STEP 6

Now, we have the simplified form of $(-2i)^{2}$ . Next, we need to subtract $(25-12i)$ from $(53 -28i)$ .
$(53 -28i) - (25-12i)$

STEP 7

istribute the negative sign to both terms in the parentheses.
$(53 -28i) -25 +12i$

STEP 8

Combine like terms.
$(53 -25) + (-28i +12i)$

STEP 9

implify the expression.
$28 -16i$ The simplified form of the given expression is $28 -16i$ .

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QuestionSimplify: $(7-2 i)^{2}-(25-12 i)$ . Choose the correct answer: a. $28+21 i$ , b. $28-16 i$ , c. $20-40 i$ , d. $20-16 i$ .

Studdy Solution

STEP 1

Assumptions1. We are dealing with complex numbers, where $i$ is the imaginary unit with the property $i^{} = -1$ . . The operations of addition, subtraction, and multiplication apply to complex numbers in the same way as they do to real numbers.

STEP 2

First, we need to simplify the square of $(7-2i)$ . We can do this by multiplying the complex number by itself.
$(7-2i)^{2} = (7-2i) \times (7-2i)$

STEP 3

Now, let's expand the expression using the distributive property of multiplication over addition.
$(7-2i) \times (7-2i) =7 \times7 +7 \times (-2i) + (-2i) \times7 + (-2i) \times (-2i)$

STEP 4

implify the expression by performing the multiplication.
$(7-2i) \times (7-2i) =49 -14i -14i +4$

STEP 5

Combine like terms.
$(7-2i) \times (7-2i) =53 -28i$

STEP 6

Now, we have the simplified form of $(-2i)^{2}$ . Next, we need to subtract $(25-12i)$ from $(53 -28i)$ .
$(53 -28i) - (25-12i)$

STEP 7

istribute the negative sign to both terms in the parentheses.
$(53 -28i) -25 +12i$

STEP 8

Combine like terms.
$(53 -25) + (-28i +12i)$

STEP 9

implify the expression.
$28 -16i$ The simplified form of the given expression is $28 -16i$ .

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QuestionSimplify: (7−2i)2−(25−12i)(7-2 i)^{2}-(25-12 i)(7−2i)2−(25−12i). Choose the correct answer: a. 28+21i28+21 i28+21i, b. 28−16i28-16 i28−16i, c. 20−40i20-40 i20−40i, d. 20−16i20-16 i20−16i.

Studdy Solution

STEP 1

Assumptions1. We are dealing with complex numbers, where iii is the imaginary unit with the property i=−1i^{} = -1i=−1. . The operations of addition, subtraction, and multiplication apply to complex numbers in the same way as they do to real numbers.

STEP 2

First, we need to simplify the square of (7−2i)(7-2i)(7−2i). We can do this by multiplying the complex number by itself.(7−2i)2=(7−2i)×(7−2i)(7-2i)^{2} = (7-2i) \times (7-2i) (7−2i)2=(7−2i)×(7−2i)

STEP 3

STEP 4

implify the expression by performing the multiplication.(7−2i)×(7−2i)=49−14i−14i+4(7-2i) \times (7-2i) =49 -14i -14i +4(7−2i)×(7−2i)=49−14i−14i+4

STEP 5

Combine like terms.(7−2i)×(7−2i)=53−28i(7-2i) \times (7-2i) =53 -28i(7−2i)×(7−2i)=53−28i

STEP 6

Now, we have the simplified form of (−2i)2(-2i)^{2}(−2i)2. Next, we need to subtract (25−12i)(25-12i)(25−12i) from (53−28i)(53 -28i)(53−28i).(53−28i)−(25−12i)(53 -28i) - (25-12i) (53−28i)−(25−12i)

STEP 7

istribute the negative sign to both terms in the parentheses.(53−28i)−25+12i(53 -28i) -25 +12i(53−28i)−25+12i

STEP 8

Combine like terms.(53−25)+(−28i+12i)(53 -25) + (-28i +12i) (53−25)+(−28i+12i)

STEP 9

implify the expression.28−16i28 -16i28−16iThe simplified form of the given expression is 28−16i28 -16i28−16i.

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QuestionSimplify: $(7-2 i)^{2}-(25-12 i)$ . Choose the correct answer: a. $28+21 i$ , b. $28-16 i$ , c. $20-40 i$ , d. $20-16 i$ .

Assumptions1. We are dealing with complex numbers, where $i$ is the imaginary unit with the property $i^{} = -1$ . . The operations of addition, subtraction, and multiplication apply to complex numbers in the same way as they do to real numbers.

First, we need to simplify the square of $(7-2i)$ . We can do this by multiplying the complex number by itself.
$(7-2i)^{2} = (7-2i) \times (7-2i)$

implify the expression by performing the multiplication.
$(7-2i) \times (7-2i) =49 -14i -14i +4$

Combine like terms.
$(7-2i) \times (7-2i) =53 -28i$

Now, we have the simplified form of $(-2i)^{2}$ . Next, we need to subtract $(25-12i)$ from $(53 -28i)$ .
$(53 -28i) - (25-12i)$

istribute the negative sign to both terms in the parentheses.
$(53 -28i) -25 +12i$

Combine like terms.
$(53 -25) + (-28i +12i)$

implify the expression.
$28 -16i$ The simplified form of the given expression is $28 -16i$ .