Not breaking down the cube root into its prime factors

Oct 13, 2025 by miodeya.com 105 views

Understanding the Problem

The given expression involves a cube root and a coefficient. To simplify the expression, we need to break down the cube root into its prime factors and then simplify the expression.

Breaking Down the Cube Root

The cube root of $648x^4y^8$ can be broken down as follows:

\sqrt[3]{648x^4y^8} = \sqrt[3]{2^3 \cdot 3^4 \cdot x^4 \cdot y^8}

Simplifying the Cube Root

Using the properties of exponents, we can simplify the cube root as follows:

\sqrt[3]{2^3 \cdot 3^4 \cdot x^4 \cdot y^8} = 2 \cdot 3 \cdot x \cdot y^2 \sqrt[3]{3^2 \cdot x^2 \cdot y^2}

Simplifying the Expression

Now, we can simplify the original expression by multiplying the coefficient with the simplified cube root:

3x \sqrt[3]{648x^4y^8} = 3x \cdot 2 \cdot 3 \cdot x \cdot y^2 \sqrt[3]{3^2 \cdot x^2 \cdot y^2}

Final Simplification

Simplifying the expression further, we get:

3x \sqrt[3]{648x^4y^8} = 18x^2y^2 \sqrt[3]{3^2 \cdot x^2 \cdot y^2}

Conclusion

The simplified expression is $18x^2y^2 \sqrt[3]{3^2 \cdot x^2 \cdot y^2}$ , which can be rewritten as $18x^2y^2 \sqrt[3]{3xy^2}$ .

Comparison with Options

Comparing the simplified expression with the given options, we can see that the correct answer is:

Option D: $18x^2y^2 \sqrt[3]{3xy^2}$

Final Answer

The final answer is Option D: $18x^2y^2 \sqrt[3]{3xy^2}$ .

Understanding the Problem

The given expression involves a cube root and a coefficient. To simplify the expression, we need to break down the cube root into its prime factors and then simplify the expression.

Q&A Session

Q: What is the first step in simplifying the expression?

A: The first step in simplifying the expression is to break down the cube root into its prime factors.

Q: How do we break down the cube root?

A: We break down the cube root by expressing the number inside the cube root as a product of its prime factors.

Q: What are the prime factors of 648?

A: The prime factors of 648 are $2^3 \cdot 3^4$ .

Q: How do we simplify the cube root?

A: We simplify the cube root by using the properties of exponents. We can rewrite the cube root as $2 \cdot 3 \cdot x \cdot y^2 \sqrt[3]{3^2 \cdot x^2 \cdot y^2}$ .

Q: What is the next step in simplifying the expression?

A: The next step in simplifying the expression is to multiply the coefficient with the simplified cube root.

Q: How do we simplify the expression further?

A: We simplify the expression further by multiplying the coefficient with the simplified cube root. This gives us $18x^2y^2 \sqrt[3]{3^2 \cdot x^2 \cdot y^2}$ .

Q: What is the final simplified expression?

A: The final simplified expression is $18x^2y^2 \sqrt[3]{3xy^2}$ .

Q: How do we compare the simplified expression with the given options?

A: We compare the simplified expression with the given options by looking at the coefficient and the cube root.

Q: Which option is the correct answer?

A: The correct answer is Option D: $18x^2y^2 \sqrt[3]{3xy^2}$ .

Common Mistakes

Not breaking down the cube root into its prime factors: This is the most common mistake when simplifying expressions with cube roots.
Not using the properties of exponents: This can lead to incorrect simplification of the cube root.
Not comparing the simplified expression with the given options: This can lead to choosing the wrong answer.

Tips and Tricks

Break down the cube root into its prime factors: This is the first step in simplifying the expression.
Use the properties of exponents: This will help you simplify the cube root correctly.
Compare the simplified expression with the given options: This will help you choose the correct answer.

Conclusion

Simplifying expressions with cube roots requires breaking down the cube root into its prime factors and then simplifying the expression using the properties of exponents. By following these steps and comparing the simplified expression with the given options, you can choose the correct answer.

Final Answer

The final answer is Option D: $18x^2y^2 \sqrt[3]{3xy^2}$ .