Solved on Dec 08, 2023

Question 5: Find the modulus of the complex number $-4+3i$ .
Question 6: Simplify the complex fraction $(2-4i)/(1+3i)$ and find the values of $a$ and $b$ where $a+bi$ is the result.

STEP 1

Assumptions for Question 5
1. The complex number is $-4 + 3i$ .
2. The modulus of a complex number $z = a + bi$ is given by $\sqrt{a^2 + b^2}$ .

STEP 2

We will calculate the modulus of the complex number $-4 + 3i$ using the formula for the modulus of a complex number.
$Modulus = \sqrt{a^2 + b^2}$

STEP 3

Substitute $a = -4$ and $b = 3$ into the formula.
$Modulus = \sqrt{(-4)^2 + (3)^2}$

STEP 4

Calculate the squares of $a$ and $b$ .
$Modulus = \sqrt{16 + 9}$

STEP 5

Add the squares to find the modulus.
$Modulus = \sqrt{25}$

STEP 6

Calculate the square root to find the modulus.
$Modulus = 5$
The modulus of the complex number $-4 + 3i$ is $5$ .

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Question 5: Find the modulus of the complex number $-4+3i$ .
Question 6: Simplify the complex fraction $(2-4i)/(1+3i)$ and find the values of $a$ and $b$ where $a+bi$ is the result.

STEP 1

Assumptions for Question 5
1. The complex number is $-4 + 3i$ .
2. The modulus of a complex number $z = a + bi$ is given by $\sqrt{a^2 + b^2}$ .

STEP 2

We will calculate the modulus of the complex number $-4 + 3i$ using the formula for the modulus of a complex number.
$Modulus = \sqrt{a^2 + b^2}$

STEP 3

Substitute $a = -4$ and $b = 3$ into the formula.
$Modulus = \sqrt{(-4)^2 + (3)^2}$

STEP 4

Calculate the squares of $a$ and $b$ .
$Modulus = \sqrt{16 + 9}$

STEP 5

Add the squares to find the modulus.
$Modulus = \sqrt{25}$

STEP 6

Calculate the square root to find the modulus.
$Modulus = 5$
The modulus of the complex number $-4 + 3i$ is $5$ .

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Question 5: Find the modulus of the complex number −4+3i-4+3i−4+3i.Question 6: Simplify the complex fraction (2−4i)/(1+3i)(2-4i)/(1+3i)(2−4i)/(1+3i) and find the values of aaa and bbb where a+bia+bia+bi is the result.

STEP 1

Assumptions for Question 51. The complex number is −4+3i-4 + 3i−4+3i.2. The modulus of a complex number z=a+biz = a + biz=a+bi is given by a2+b2\sqrt{a^2 + b^2}a2+b2​.

STEP 2

We will calculate the modulus of the complex number −4+3i-4 + 3i−4+3i using the formula for the modulus of a complex number.Modulus=a2+b2Modulus = \sqrt{a^2 + b^2}Modulus=a2+b2​

STEP 3

Substitute a=−4a = -4a=−4 and b=3b = 3b=3 into the formula.Modulus=(−4)2+(3)2Modulus = \sqrt{(-4)^2 + (3)^2}Modulus=(−4)2+(3)2​

STEP 4

Calculate the squares of aaa and bbb.Modulus=16+9Modulus = \sqrt{16 + 9}Modulus=16+9​

STEP 5

Add the squares to find the modulus.Modulus=25Modulus = \sqrt{25}Modulus=25​

STEP 6

Calculate the square root to find the modulus.Modulus=5Modulus = 5Modulus=5The modulus of the complex number −4+3i-4 + 3i−4+3i is 555.

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Question 5: Find the modulus of the complex number $-4+3i$ .
Question 6: Simplify the complex fraction $(2-4i)/(1+3i)$ and find the values of $a$ and $b$ where $a+bi$ is the result.

Assumptions for Question 5
1. The complex number is $-4 + 3i$ .
2. The modulus of a complex number $z = a + bi$ is given by $\sqrt{a^2 + b^2}$ .

We will calculate the modulus of the complex number $-4 + 3i$ using the formula for the modulus of a complex number.
$Modulus = \sqrt{a^2 + b^2}$

Substitute $a = -4$ and $b = 3$ into the formula.
$Modulus = \sqrt{(-4)^2 + (3)^2}$

Calculate the squares of $a$ and $b$ .
$Modulus = \sqrt{16 + 9}$

Add the squares to find the modulus.
$Modulus = \sqrt{25}$

Calculate the square root to find the modulus.
$Modulus = 5$
The modulus of the complex number $-4 + 3i$ is $5$ .