Solve For \[$ Z \$\] In The Equation \[$ Z^3 = -0.027 \$\].One Or More Solutions, \[$ Z = \$\] \[$\square\$\]No Solution

Introduction

Solving for $z$ in the equation $z^3 = -0.027$ involves finding the cube root of $-0.027$. This equation is a simple cubic equation, and we will use the properties of cube roots to find the solution.

Understanding Cube Roots

A cube root of a number is a value that, when multiplied by itself twice, gives the original number. In mathematical notation, if $x$ is a cube root of $y$, then $x^3 = y$. The cube root of a negative number is also a negative number.

Finding the Cube Root of $-0.027$

To find the cube root of $-0.027$, we can use a calculator or a mathematical software package. Alternatively, we can use the fact that the cube root of a negative number is also a negative number.

Using a Calculator or Mathematical Software

Using a calculator or mathematical software package, we can find the cube root of $-0.027$ as follows:

$$z = \sqrt[3]{-0.027} \approx -0.303$$

Using the Properties of Cube Roots

Alternatively, we can use the properties of cube roots to find the solution. Since $z^3 = -0.027$, we can take the cube root of both sides of the equation to get:

$$z = \sqrt[3]{-0.027}$$

Simplifying the Expression

We can simplify the expression by using the fact that the cube root of a negative number is also a negative number. Therefore, we can write:

$$z = -\sqrt[3]{0.027}$$

Evaluating the Expression

Evaluating the expression, we get:

$$z \approx -0.303$$

Conclusion

In conclusion, the solution to the equation $z^3 = -0.027$ is $z \approx -0.303$. This solution is a negative number, as expected.

Additional Solutions

In addition to the solution $z \approx -0.303$, there may be other solutions to the equation $z^3 = -0.027$. These solutions are complex numbers, and they can be found using the properties of complex numbers.

Complex Solutions

The complex solutions to the equation $z^3 = -0.027$ can be found using the following formula:

$$z = -0.303 + ki$$

where $k$ is a real number, and $i$ is the imaginary unit.

Evaluating the Complex Solutions

Evaluating the complex solutions, we get:

$$z \approx -0.303 + 0.303i$$

$$z \approx -0.303 - 0.303i$$

$$z \approx 0.303 + 0.303i$$

$$z \approx 0.303 - 0.303i$$

Conclusion

In conclusion, the solutions to the equation $z^3 = -0.027$ are $z \approx -0.303$, $z \approx -0.303 + 0.303i$, $z \approx -0.303 - 0.303i$, $z \approx 0.303 + 0.303i$, and $z \approx 0.303 - 0.303i$.

Final Answer

The final answer is: $\boxed{-0.303}$