The Cube Root Shown Has Been Written As A Product Where One Of The Factors Is A Perfect Cube:${ \sqrt[3]{162}=\sqrt[3]{27} \cdot \sqrt[3]{6} }$The Simplest Form Of { \sqrt[3]{162}$}$ Is { A \sqrt[3]{b}$} , W H E R E : \[ , Where:\[ , W H Ere : \[ A

Introduction

In mathematics, the cube root of a number is a value that, when multiplied by itself twice, gives the original number. The cube root is denoted by the radical symbol $\sqrt[3]{x}$, where $x$ is the number inside the radical. In this article, we will explore the concept of writing the cube root as a product where one of the factors is a perfect cube. We will use the given example, $\sqrt[3]{162}=\sqrt[3]{27} \cdot \sqrt[3]{6}$, to illustrate this concept and find the simplest form of $\sqrt[3]{162}$.

Breaking Down the Cube Root

To understand how the cube root can be written as a product, let's break down the given example. We have $\sqrt[3]{162}=\sqrt[3]{27} \cdot \sqrt[3]{6}$. Here, $\sqrt[3]{27}$ is a perfect cube, as $27=3^3$. This means that $\sqrt[3]{27}=3$, as the cube root of a perfect cube is the number that, when multiplied by itself twice, gives the original number.

Simplifying the Cube Root

Now, let's simplify the cube root $\sqrt[3]{162}$. We can start by factoring $162$ into its prime factors. We have $162=2 \cdot 3^4 \cdot 3$. Since $\sqrt[3]{27}=3$, we can rewrite $\sqrt[3]{162}$ as $\sqrt[3]{2 \cdot 3^4 \cdot 3}$. We can then simplify this expression by taking out the perfect cube factor, $3^3$, which is equal to $27$. This gives us $\sqrt[3]{162}=\sqrt[3]{27} \cdot \sqrt[3]{2 \cdot 3}$.

Finding the Simplest Form

Now that we have simplified the cube root, we can find the simplest form of $\sqrt[3]{162}$. We have $\sqrt[3]{162}=\sqrt[3]{27} \cdot \sqrt[3]{2 \cdot 3}$. Since $\sqrt[3]{27}=3$, we can rewrite this expression as $3 \cdot \sqrt[3]{2 \cdot 3}$. This is the simplest form of $\sqrt[3]{162}$.

Conclusion

In this article, we have explored the concept of writing the cube root as a product where one of the factors is a perfect cube. We used the given example, $\sqrt[3]{162}=\sqrt[3]{27} \cdot \sqrt[3]{6}$, to illustrate this concept and find the simplest form of $\sqrt[3]{162}$. We broke down the cube root, simplified it, and found the simplest form of $\sqrt[3]{162}$, which is $3 \cdot \sqrt[3]{2 \cdot 3}$.

The Final Answer

The final answer is $\boxed{3 \sqrt[3]{2 \cdot 3}}$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Break down the cube root $\sqrt[3]{162}$ into its prime factors.
2. Factor $162$ into its prime factors: $162=2 \cdot 3^4 \cdot 3$.
3. Take out the perfect cube factor, $3^3$, which is equal to $27$.
4. Simplify the expression: $\sqrt[3]{162}=\sqrt[3]{27} \cdot \sqrt[3]{2 \cdot 3}$.
5. Rewrite the expression using the fact that $\sqrt[3]{27}=3$: $3 \cdot \sqrt[3]{2 \cdot 3}$.

Frequently Asked Questions

• What is the cube root of $162$?
• How can the cube root be written as a product where one of the factors is a perfect cube?
• What is the simplest form of $\sqrt[3]{162}$?

Answer to Frequently Asked Questions

• The cube root of $162$ is $\sqrt[3]{162}$.
• The cube root can be written as a product where one of the factors is a perfect cube by factoring the number inside the radical into its prime factors and taking out the perfect cube factor.
• The simplest form of $\sqrt[3]{162}$ is $3 \cdot \sqrt[3]{2 \cdot 3}$.

Further Reading

For further reading on the topic of cube roots and perfect cubes, we recommend the following resources:

• "Cube Roots and Perfect Cubes" by Math Open Reference
• "Cube Root" by Wolfram MathWorld
• "Perfect Cube" by Math Is Fun

References

• "Mathematics for the Nonmathematician" by Morris Kline
• "Calculus" by Michael Spivak
• "Algebra" by Michael Artin
  

Introduction

In our previous article, we explored the concept of writing the cube root as a product where one of the factors is a perfect cube. We used the given example, $\sqrt[3]{162}=\sqrt[3]{27} \cdot \sqrt[3]{6}$, to illustrate this concept and find the simplest form of $\sqrt[3]{162}$. In this article, we will answer some of the most frequently asked questions about cube roots and perfect cubes.

Q&A

Q: What is the cube root of a number?

A: The cube root of a number is a value that, when multiplied by itself twice, gives the original number. It is denoted by the radical symbol $\sqrt[3]{x}$, where $x$ is the number inside the radical.

Q: What is a perfect cube?

A: A perfect cube is a number that can be expressed as the cube of an integer. For example, $27$ is a perfect cube because it can be expressed as $3^3$.

Q: How can the cube root be written as a product where one of the factors is a perfect cube?

A: The cube root can be written as a product where one of the factors is a perfect cube by factoring the number inside the radical into its prime factors and taking out the perfect cube factor.

Q: What is the simplest form of $\sqrt[3]{162}$?

A: The simplest form of $\sqrt[3]{162}$ is $3 \cdot \sqrt[3]{2 \cdot 3}$.

Q: How do I find the cube root of a number?

A: To find the cube root of a number, you can use a calculator or a computer program. Alternatively, you can use the formula $\sqrt[3]{x} = \sqrt[3]{a} \cdot \sqrt[3]{b}$, where $x = a \cdot b$ and $a$ and $b$ are integers.

Q: What is the difference between a cube root and a square root?

A: A cube root is a value that, when multiplied by itself twice, gives the original number, while a square root is a value that, when multiplied by itself, gives the original number.

Q: Can a cube root be a perfect square?

A: No, a cube root cannot be a perfect square. A perfect square is a number that can be expressed as the square of an integer, while a cube root is a value that, when multiplied by itself twice, gives the original number.

Q: How do I simplify a cube root expression?

A: To simplify a cube root expression, you can factor the number inside the radical into its prime factors and take out the perfect cube factor.

Conclusion

In this article, we have answered some of the most frequently asked questions about cube roots and perfect cubes. We have explained the concept of writing the cube root as a product where one of the factors is a perfect cube and provided examples of how to simplify cube root expressions.

Further Reading

For further reading on the topic of cube roots and perfect cubes, we recommend the following resources:

• "Cube Roots and Perfect Cubes" by Math Open Reference
• "Cube Root" by Wolfram MathWorld
• "Perfect Cube" by Math Is Fun

References

• "Mathematics for the Nonmathematician" by Morris Kline
• "Calculus" by Michael Spivak
• "Algebra" by Michael Artin

Related Articles

• "The Cube Root Shown Has Been Written as a Product Where One of the Factors is a Perfect Cube"
• "Simplifying Cube Root Expressions"
• "Cube Roots and Perfect Cubes: A Tutorial"

Tags

• cube root
• perfect cube
• simplifying cube root expressions
• cube roots and perfect cubes
• mathematics
• algebra
• calculus