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Consider the equation:

z^(4)=81
One solution is
z=3, another is -3 , and there are 2 more distinct complex solutions.
What are those solutions?
Choose
2 answers:
A
z=3i
B
z=27 i
c.
z=-3i
D
z=-27 i

Consider the equation: $\newline$ $z^{4}=81$ $\newline$ One solution is $z=3$ , another is $-3$ , and there are $2$ more distinct complex solutions. $\newline$ What are those solutions? $\newline$ Choose $2$ answers: $\newline$ A) $z=3 i$ $\newline$ B) $z=27 i$ $\newline$ C) $z=-3 i$ $\newline$ D) $z=-27 i$

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Solve rational equations

Full solution

Q. Consider the equation:

\newline

z^{4}=81

\newline

One solution is

z=3

, another is

-3

, and there are

2

more distinct complex solutions.

\newline

What are those solutions?

\newline

Choose

2

answers:

\newline

A)

z=3 i

\newline

B)

z=27 i

\newline

C)

z=-3 i

\newline

D)

z=-27 i

Rewrite as Difference of Squares: Recognize that the equation $z^{4}=81$ can be rewritten as $z^{4} - 81 = 0$ , which is a difference of squares.
Factor using Formula: Factor the equation using the difference of squares formula, which states that $a^2 - b^2 = (a + b)(a - b)$ . In this case, we have $z^{4} - 81 = (z^2 + 9)(z^2 - 9)$ .
Further Factorization: Notice that $z^2 - 9$ is also a difference of squares, so we can factor it further to $(z + 3)(z - 3)$ . Now we have $(z^2 + 9)(z + 3)(z - 3) = 0$ .
Find Solutions for $z^2 + 9$ : We already know that $z = 3$ and $z = -3$ are solutions from the factors $(z - 3)$ and $(z + 3)$ . Now we need to find the solutions for $z^2 + 9 = 0$ .
Solve for $z$ : Solve the equation $z^2 + 9 = 0$ for $z$ . Subtract $9$ from both sides to get $z^2 = -9$ .
Complex Solutions: Take the square root of both sides to solve for $z$ . The square root of $-9$ is $3i$ and $-3i$ , so $z = 3i$ and $z = -3i$ are the complex solutions.
Check Validity: Check the solutions by substituting them back into the original equation $z^{4}=81$ . For $z = 3i$ , $(3i)^{4} = 81i^{4} = 81(-1)^{2} = 81$ , which is true. For $z = -3i$ , $(-3i)^{4} = 81i^{4} = 81(-1)^{2} = 81$ , which is also true.

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