Reformat The Expression Below:${ \begin{aligned} &=\sqrt[3]{-4^3} \ &=-4 \ &= \ &j \cdot \sqrt[3]{81} \end{aligned} }$

Introduction

When dealing with cube roots of negative numbers, it's essential to understand the properties of radicals and how they interact with exponents. In this article, we'll explore the given expression and simplify it step by step, using the properties of cube roots and exponents to arrive at the final result.

Understanding the Given Expression

The given expression is:

$$\begin{aligned} &=\sqrt[3]{-4^3} \\ &=-4 \\ &= \\ &j \cdot \sqrt[3]{81} \end{aligned}$$

This expression involves a cube root of a negative number, which can be simplified using the properties of radicals and exponents.

Simplifying the Cube Root of a Negative Number

To simplify the cube root of a negative number, we can use the property that $\sqrt[3]{-a} = -\sqrt[3]{a}$ for any positive real number $a$. Applying this property to the given expression, we get:

$$\begin{aligned} \sqrt[3]{-4^3} &= -\sqrt[3]{4^3} \\ &= -4^{\frac{3}{3}} \\ &= -4 \end{aligned}$$

Understanding the Relationship Between Cube Roots and Exponents

The cube root of a number $a$ can be expressed as $a^{\frac{1}{3}}$. In the given expression, we have $-4^3$, which can be rewritten as $(-4)^3$. Using the property of exponents, we can rewrite this as $-4^3 = (-4)^3 = (-1)^3 \cdot 4^3 = -4^3$.

Simplifying the Expression Further

Now that we have simplified the cube root of the negative number, we can focus on simplifying the expression further. We have:

$$\begin{aligned} &=-4 \\ &= \\ &j \cdot \sqrt[3]{81} \end{aligned}$$

To simplify this expression, we can use the property that $\sqrt[3]{a^3} = a$ for any positive real number $a$. Applying this property to the given expression, we get:

$$\begin{aligned} j \cdot \sqrt[3]{81} &= j \cdot 81^{\frac{1}{3}} \\ &= j \cdot 9 \\ &= 9j \end{aligned}$$

Combining the Simplified Expressions

Now that we have simplified both parts of the expression, we can combine them to get the final result. We have:

$$\begin{aligned} &=-4 \\ &= 9j \end{aligned}$$

Conclusion

In this article, we simplified the given expression step by step, using the properties of radicals and exponents to arrive at the final result. We started by simplifying the cube root of a negative number, then used the property of exponents to rewrite the expression, and finally combined the simplified expressions to get the final result.

Final Answer

The final answer is $\boxed{-4 + 9j}$.

Discussion

The given expression involves a cube root of a negative number, which can be simplified using the properties of radicals and exponents. The final result is a complex number, which can be expressed in the form $a + bj$, where $a$ and $b$ are real numbers and $j$ is the imaginary unit.

Related Topics

• Simplifying cube roots of negative numbers
• Properties of radicals and exponents
• Complex numbers

References

• [1] "Radicals and Exponents" by [Author]
• [2] "Complex Numbers" by [Author]

Keywords

• Cube root of a negative number
• Properties of radicals and exponents
• Complex numbers
• Simplifying expressions
• Imaginary unit
  

Introduction

In our previous article, we simplified the given expression step by step, using the properties of radicals and exponents to arrive at the final result. In this article, we'll answer some frequently asked questions related to the topic, providing additional insights and clarifications.

Q&A

Q: What is the cube root of a negative number?

A: The cube root of a negative number can be simplified using the property that $\sqrt[3]{-a} = -\sqrt[3]{a}$ for any positive real number $a$. This means that the cube root of a negative number is equal to the negative of the cube root of its absolute value.

Q: How do you simplify the cube root of a negative number?

A: To simplify the cube root of a negative number, you can use the property mentioned above. For example, $\sqrt[3]{-4^3} = -\sqrt[3]{4^3} = -4^{\frac{3}{3}} = -4$.

Q: What is the relationship between cube roots and exponents?

A: The cube root of a number $a$ can be expressed as $a^{\frac{1}{3}}$. This means that the cube root of a number is equal to the number raised to the power of $\frac{1}{3}$.

Q: How do you simplify the expression $j \cdot \sqrt[3]{81}$?

A: To simplify this expression, you can use the property that $\sqrt[3]{a^3} = a$ for any positive real number $a$. Applying this property to the given expression, we get $j \cdot \sqrt[3]{81} = j \cdot 81^{\frac{1}{3}} = j \cdot 9 = 9j$.

Q: What is the final result of the given expression?

A: The final result of the given expression is $\boxed{-4 + 9j}$.

Q: What are some related topics to this article?

A: Some related topics to this article include simplifying cube roots of negative numbers, properties of radicals and exponents, and complex numbers.

Q: What are some references for further reading?

A: Some references for further reading on this topic include [1] "Radicals and Exponents" by [Author] and [2] "Complex Numbers" by [Author].

Additional Insights

• The cube root of a negative number can be simplified using the property that $\sqrt[3]{-a} = -\sqrt[3]{a}$ for any positive real number $a$.
• The cube root of a number $a$ can be expressed as $a^{\frac{1}{3}}$.
• The expression $j \cdot \sqrt[3]{81}$ can be simplified using the property that $\sqrt[3]{a^3} = a$ for any positive real number $a$.
• The final result of the given expression is $\boxed{-4 + 9j}$.

Conclusion

In this article, we answered some frequently asked questions related to the topic of simplifying the cube root of a negative number. We provided additional insights and clarifications, and discussed some related topics and references for further reading.

Final Answer

The final answer is $\boxed{-4 + 9j}$.

Discussion

The given expression involves a cube root of a negative number, which can be simplified using the properties of radicals and exponents. The final result is a complex number, which can be expressed in the form $a + bj$, where $a$ and $b$ are real numbers and $j$ is the imaginary unit.

Related Topics

• Simplifying cube roots of negative numbers
• Properties of radicals and exponents
• Complex numbers

References

• [1] "Radicals and Exponents" by [Author]
• [2] "Complex Numbers" by [Author]

Keywords

• Cube root of a negative number
• Properties of radicals and exponents
• Complex numbers
• Simplifying expressions
• Imaginary unit