What Is The Radical Part Of Both $\sqrt[3]{54}$ And $\sqrt[3]{128}$ When The Expressions Are Simplified?A. \$\sqrt[3]{2}$[/tex\] B. $\sqrt{2}$ C. $\sqrt[3]{3}$ D. 2

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What is the Radical Part of Both $\sqrt[3]{54}$ and $\sqrt[3]{128}$ When the Expressions are Simplified?

Understanding the Concept of Radical Expressions

Radical expressions are mathematical expressions that involve the use of radicals, which are the roots of numbers. In this case, we are dealing with cube roots, denoted by the symbol $\sqrt[3]{x}$. The cube root of a number is a value that, when multiplied by itself twice, gives the original number. For example, the cube root of 27 is 3, because 3 multiplied by itself twice gives 27.

Breaking Down the Given Expressions

We are given two radical expressions: $\sqrt[3]{54}$ and $\sqrt[3]{128}$. To simplify these expressions, we need to find the prime factorization of the numbers inside the cube root.

Prime Factorization of 54

The prime factorization of 54 is:

54 = 2 × 3 × 3 × 3

Prime Factorization of 128

The prime factorization of 128 is:

128 = 2 × 2 × 2 × 2 × 2 × 2 × 2

Simplifying the Radical Expressions

Now that we have the prime factorization of both numbers, we can simplify the radical expressions.

For $\sqrt[3]{54}$, we can take out the perfect cube factors, which are 3 × 3 × 3. This leaves us with:

\sqrt[3]{54}$ = $\sqrt[3]{2 × 3 × 3 × 3}

= $\sqrt[3]{3 × 3 × 3}$ × $\sqrt[3]{2}$

= 3 × $\sqrt[3]{2}$

For $\sqrt[3]{128}$, we can take out the perfect cube factors, which are 2 × 2 × 2. This leaves us with:

\sqrt[3]{128}$ = $\sqrt[3]{2 × 2 × 2 × 2 × 2 × 2 × 2}

= $\sqrt[3]{2 × 2 × 2}$ × $\sqrt[3]{2 × 2 × 2}$ × $\sqrt[3]{2}$

= 2 × 2 × $\sqrt[3]{2}$

= 4 × $\sqrt[3]{2}$

Identifying the Radical Part

Now that we have simplified the radical expressions, we can identify the radical part of both expressions.

For $\sqrt[3]{54}$, the radical part is $\sqrt[3]{2}$.

For $\sqrt[3]{128}$, the radical part is also $\sqrt[3]{2}$.

Conclusion

In conclusion, the radical part of both $\sqrt[3]{54}$ and $\sqrt[3]{128}$ when the expressions are simplified is $\sqrt[3]{2}$.

Answer

The correct answer is A. $\sqrt[3]{2}$.

Additional Tips and Tricks

  • When simplifying radical expressions, it's essential to identify the perfect cube factors and take them out.
  • The radical part of a radical expression is the part that remains after taking out the perfect cube factors.
  • In this case, both radical expressions have the same radical part, which is $\sqrt[3]{2}$.

Common Mistakes to Avoid

  • Not identifying the perfect cube factors when simplifying radical expressions.
  • Not taking out the perfect cube factors when simplifying radical expressions.
  • Not identifying the radical part of a radical expression.

Real-World Applications

  • Radical expressions are used in various real-world applications, such as physics, engineering, and computer science.
  • Understanding how to simplify radical expressions is essential in these fields.

Practice Problems

  • Simplify the radical expression $\sqrt[3]{216}$.
  • Simplify the radical expression $\sqrt[3]{243}$.

Answer Key

  • \sqrt[3]{216}$ = 6

  • \sqrt[3]{243}$ = 3 × 3<br/>

Q&A: Simplifying Radical Expressions

Frequently Asked Questions

Q: What is the difference between a radical expression and a rational expression?

A: A radical expression is a mathematical expression that involves the use of radicals, which are the roots of numbers. A rational expression, on the other hand, is a mathematical expression that involves the use of fractions.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to identify the perfect cube factors and take them out. This involves finding the prime factorization of the number inside the radical and taking out the perfect cube factors.

Q: What is the radical part of a radical expression?

A: The radical part of a radical expression is the part that remains after taking out the perfect cube factors. It is the part that is left inside the radical.

Q: Can I simplify a radical expression with a negative number inside the radical?

A: Yes, you can simplify a radical expression with a negative number inside the radical. However, you need to remember that the cube root of a negative number is also negative.

Q: How do I simplify a radical expression with a variable inside the radical?

A: To simplify a radical expression with a variable inside the radical, you need to identify the perfect cube factors of the variable and take them out. This involves finding the prime factorization of the variable and taking out the perfect cube factors.

Q: Can I simplify a radical expression with a fraction inside the radical?

A: Yes, you can simplify a radical expression with a fraction inside the radical. However, you need to remember that the cube root of a fraction is also a fraction.

Q: How do I simplify a radical expression with multiple radicals inside the radical?

A: To simplify a radical expression with multiple radicals inside the radical, you need to identify the perfect cube factors of each radical and take them out. This involves finding the prime factorization of each radical and taking out the perfect cube factors.

Q: Can I simplify a radical expression with a radical inside the radical?

A: Yes, you can simplify a radical expression with a radical inside the radical. However, you need to remember that the cube root of a radical is also a radical.

Q: How do I simplify a radical expression with a negative number and a variable inside the radical?

A: To simplify a radical expression with a negative number and a variable inside the radical, you need to identify the perfect cube factors of the variable and take them out. This involves finding the prime factorization of the variable and taking out the perfect cube factors.

Q: Can I simplify a radical expression with a fraction and a variable inside the radical?

A: Yes, you can simplify a radical expression with a fraction and a variable inside the radical. However, you need to remember that the cube root of a fraction is also a fraction.

Q: How do I simplify a radical expression with multiple radicals and variables inside the radical?

A: To simplify a radical expression with multiple radicals and variables inside the radical, you need to identify the perfect cube factors of each radical and take them out. This involves finding the prime factorization of each radical and taking out the perfect cube factors.

Q: Can I simplify a radical expression with a radical and a variable inside the radical?

A: Yes, you can simplify a radical expression with a radical and a variable inside the radical. However, you need to remember that the cube root of a radical is also a radical.

Conclusion

Simplifying radical expressions is an essential skill in mathematics, and it has many real-world applications. By understanding how to simplify radical expressions, you can solve complex mathematical problems and apply mathematical concepts to real-world situations.

Practice Problems

  • Simplify the radical expression $\sqrt[3]{216}$.
  • Simplify the radical expression $\sqrt[3]{243}$.
  • Simplify the radical expression $\sqrt[3]{64}$.
  • Simplify the radical expression $\sqrt[3]{125}$.

Answer Key

  • \sqrt[3]{216}$ = 6

  • \sqrt[3]{243}$ = 3 × 3

  • \sqrt[3]{64}$ = 4

  • \sqrt[3]{125}$ = 5