Question Find the cube of $(-6+5i)$ in the form $a+bi$ .

Studdy Solution

STEP 1

Assumptions
1. We are given a complex number $(-6+5i)$ .
2. We need to calculate the cube of this complex number.
3. The result should be in the form $a+bi$ where $a$ and $b$ are real numbers.

STEP 2

To find the cube of the complex number $(-6+5i)$ , we will use the formula for the cube of a binomial $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ .

STEP 3

Identify $a$ and $b$ from the complex number $(-6+5i)$ where $a=-6$ and $b=5i$ .

STEP 4

Calculate $a^3$ using the value of $a$ .
$a^3 = (-6)^3 = -216$

STEP 5

Calculate $3a^2b$ using the values of $a$ and $b$ .
$3a^2b = 3(-6)^2(5i) = 3 \cdot 36 \cdot 5i = 540i$

STEP 6

Calculate $3ab^2$ using the values of $a$ and $b$ . Remember that $i^2 = -1$ .
$3ab^2 = 3(-6)(5i)^2 = 3(-6)(25)(-1) = 450$

STEP 7

Calculate $b^3$ using the value of $b$ . Remember to apply $i^2 = -1$ when necessary.
$b^3 = (5i)^3 = 5^3i^3 = 125i^3 = 125i^2 \cdot i = 125(-1)i = -125i$

STEP 8

Now, add all the terms together to get the cube of the complex number.
$(-6+5i)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

STEP 9

Substitute the calculated values into the equation.
$(-6+5i)^3 = -216 + 540i + 450 - 125i$

STEP 10

Combine like terms to get the result in $a+bi$ form.
$(-6+5i)^3 = (-216 + 450) + (540i - 125i)$

STEP 11

Perform the addition to find the real part $a$ and the imaginary part $b$ .
$a = -216 + 450 = 234$ $b = 540i - 125i = 415i$

STEP 12

Write the final answer in $a+bi$ form.
$(-6+5i)^3 = 234 + 415i$
The cube of the complex number $(-6+5i)$ is $234+415i$ .

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