QuestionCalculate the product of $3 + 5i$ and its conjugate, then express the result in the form $a + bi$ .

Studdy Solution

STEP 1

Assumptions1. The given complex number is $3+5i$ . The conjugate of a complex number $a+bi$ is $a-bi$

STEP 2

First, we need to find the conjugate of the given complex number. The conjugate of a complex number $a+bi$ is $a-bi$ .
$Conjugate = a - bi$

STEP 3

Now, plug in the given values for $a$ and $b$ to find the conjugate of the given complex number.
$Conjugate =3 -5i$

STEP 4

Now that we have the conjugate of the given complex number, we can find the product of the complex number and its conjugate. The product of a complex number $a+bi$ and its conjugate $a-bi$ is $(a+bi)(a-bi)$ .
$Product = (a+bi)(a-bi)$

STEP 5

Plug in the values for $a$ and $b$ to find the product of the complex number and its conjugate.
$Product = (3+5i)(3-5i)$

STEP 6

Expand the product using the distributive property.
$Product =3(3) +3(-5i) +5i(3) +5i(-5i)$

STEP 7

implify the product.
$Product =9 -15i +15i -25i^2$

STEP 8

Remember that $i^2$ is equal to $-1$ .
$i^2 = -1$ $Product = -15i +15i +25$

STEP 9

Combine like terms.
$Product =34$ The product of $3+5i$ and its conjugate is $34$ .

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QuestionCalculate the product of $3 + 5i$ and its conjugate, then express the result in the form $a + bi$ .

Studdy Solution

STEP 1

Assumptions1. The given complex number is $3+5i$ . The conjugate of a complex number $a+bi$ is $a-bi$

STEP 2

First, we need to find the conjugate of the given complex number. The conjugate of a complex number $a+bi$ is $a-bi$ .
$Conjugate = a - bi$

STEP 3

Now, plug in the given values for $a$ and $b$ to find the conjugate of the given complex number.
$Conjugate =3 -5i$

STEP 4

Now that we have the conjugate of the given complex number, we can find the product of the complex number and its conjugate. The product of a complex number $a+bi$ and its conjugate $a-bi$ is $(a+bi)(a-bi)$ .
$Product = (a+bi)(a-bi)$

STEP 5

Plug in the values for $a$ and $b$ to find the product of the complex number and its conjugate.
$Product = (3+5i)(3-5i)$

STEP 6

Expand the product using the distributive property.
$Product =3(3) +3(-5i) +5i(3) +5i(-5i)$

STEP 7

implify the product.
$Product =9 -15i +15i -25i^2$

STEP 8

Remember that $i^2$ is equal to $-1$ .
$i^2 = -1$ $Product = -15i +15i +25$

STEP 9

Combine like terms.
$Product =34$ The product of $3+5i$ and its conjugate is $34$ .

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QuestionCalculate the product of 3+5i3 + 5i3+5i and its conjugate, then express the result in the form a+bia + bia+bi.

Studdy Solution

STEP 1

Assumptions1. The given complex number is 3+5i3+5i3+5i . The conjugate of a complex number a+bia+bia+bi is a−bia-bia−bi

STEP 2

First, we need to find the conjugate of the given complex number. The conjugate of a complex number a+bia+bia+bi is a−bia-bia−bi.Conjugate=a−biConjugate = a - biConjugate=a−bi

STEP 3

Now, plug in the given values for aaa and bbb to find the conjugate of the given complex number.Conjugate=3−5iConjugate =3 -5iConjugate=3−5i

STEP 4

STEP 5

Plug in the values for aaa and bbb to find the product of the complex number and its conjugate.Product=(3+5i)(3−5i)Product = (3+5i)(3-5i)Product=(3+5i)(3−5i)

STEP 6

Expand the product using the distributive property.Product=3(3)+3(−5i)+5i(3)+5i(−5i)Product =3(3) +3(-5i) +5i(3) +5i(-5i)Product=3(3)+3(−5i)+5i(3)+5i(−5i)

STEP 7

implify the product.Product=9−15i+15i−25i2Product =9 -15i +15i -25i^2Product=9−15i+15i−25i2

STEP 8

Remember that i2i^2i2 is equal to −1-1−1.i2=−1i^2 = -1i2=−1Product=−15i+15i+25Product = -15i +15i +25Product=−15i+15i+25

STEP 9

Combine like terms.Product=34Product =34Product=34The product of 3+5i3+5i3+5i and its conjugate is 343434.

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QuestionCalculate the product of $3 + 5i$ and its conjugate, then express the result in the form $a + bi$ .

Assumptions1. The given complex number is $3+5i$ . The conjugate of a complex number $a+bi$ is $a-bi$

First, we need to find the conjugate of the given complex number. The conjugate of a complex number $a+bi$ is $a-bi$ .
$Conjugate = a - bi$

Now, plug in the given values for $a$ and $b$ to find the conjugate of the given complex number.
$Conjugate =3 -5i$

Plug in the values for $a$ and $b$ to find the product of the complex number and its conjugate.
$Product = (3+5i)(3-5i)$

Expand the product using the distributive property.
$Product =3(3) +3(-5i) +5i(3) +5i(-5i)$

implify the product.
$Product =9 -15i +15i -25i^2$

Remember that $i^2$ is equal to $-1$ .
$i^2 = -1$ $Product = -15i +15i +25$

Combine like terms.
$Product =34$ The product of $3+5i$ and its conjugate is $34$ .