Simplify $\sqrt[3]{\frac{x^3 Y^{12}}{64}}$ Completely.A. $\frac{\left|x Y^3\right|}{4}$B. $\frac{x Y^4}{4}$C. $\frac{x Y^3}{64}$D. $\frac{|x| Y^4}{64}$

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will focus on simplifying the expression x3y12643\sqrt[3]{\frac{x^3 y^{12}}{64}} completely. We will break down the problem into manageable steps, and by the end of this article, you will have a clear understanding of how to simplify radical expressions.

Understanding the Problem

The given expression is x3y12643\sqrt[3]{\frac{x^3 y^{12}}{64}}. To simplify this expression, we need to understand the properties of radicals and exponents. The expression inside the radical is a fraction, and we need to simplify it before we can simplify the radical.

Step 1: Simplify the Fraction

The first step is to simplify the fraction inside the radical. We can do this by dividing the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of x3x^3 and 6464 is x3x^3, and the GCD of y12y^{12} and 6464 is y3y^3. Therefore, we can simplify the fraction as follows:

x3y1264=x3y3y964=x3y964\frac{x^3 y^{12}}{64} = \frac{x^3 y^3 y^9}{64} = \frac{x^3 y^9}{64}

Step 2: Simplify the Radical

Now that we have simplified the fraction, we can simplify the radical. To simplify a radical, we need to find the cube root of the expression inside the radical. In this case, we need to find the cube root of x3y964\frac{x^3 y^9}{64}.

x3y9643=x3y93643\sqrt[3]{\frac{x^3 y^9}{64}} = \frac{\sqrt[3]{x^3 y^9}}{\sqrt[3]{64}}

Step 3: Simplify the Cube Root

The next step is to simplify the cube root. We can do this by using the property of cube roots that states a33=a\sqrt[3]{a^3} = a. In this case, we have:

x3y93=xy3\sqrt[3]{x^3 y^9} = x y^3

Step 4: Simplify the Fraction

Now that we have simplified the cube root, we can simplify the fraction. We can do this by dividing the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of xy3x y^3 and 6464 is xy3x y^3, and the GCD of 11 and 6464 is 11. Therefore, we can simplify the fraction as follows:

xy3643=xy34\frac{x y^3}{\sqrt[3]{64}} = \frac{x y^3}{4}

Conclusion

In conclusion, we have simplified the expression x3y12643\sqrt[3]{\frac{x^3 y^{12}}{64}} completely. We broke down the problem into manageable steps, and by the end of this article, you should have a clear understanding of how to simplify radical expressions.

Answer

The final answer is xy34\boxed{\frac{x y^3}{4}}.

Discussion

This problem requires a good understanding of radical expressions and exponents. The key to simplifying this expression is to simplify the fraction inside the radical and then simplify the cube root. By following the steps outlined in this article, you should be able to simplify radical expressions with ease.

Common Mistakes

When simplifying radical expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not simplifying the fraction inside the radical
  • Not simplifying the cube root
  • Not using the property of cube roots that states a33=a\sqrt[3]{a^3} = a

Tips and Tricks

Here are some tips and tricks to help you simplify radical expressions:

  • Always simplify the fraction inside the radical before simplifying the cube root
  • Use the property of cube roots that states a33=a\sqrt[3]{a^3} = a
  • Check your work by plugging in values for xx and yy

Practice Problems

Here are some practice problems to help you practice simplifying radical expressions:

  • Simplify x3y12273\sqrt[3]{\frac{x^3 y^{12}}{27}}
  • Simplify x3y121253\sqrt[3]{\frac{x^3 y^{12}}{125}}
  • Simplify x3y122163\sqrt[3]{\frac{x^3 y^{12}}{216}}

Conclusion

Introduction

In our previous article, we discussed how to simplify radical expressions. However, we know that practice makes perfect, and there's no better way to practice than by answering questions. In this article, we will provide a Q&A guide to help you practice simplifying radical expressions.

Q1: What is the first step in simplifying a radical expression?

A1: The first step in simplifying a radical expression is to simplify the fraction inside the radical. This involves dividing the numerator and denominator by their greatest common divisor (GCD).

Q2: How do I simplify a cube root?

A2: To simplify a cube root, you need to find the cube root of the expression inside the radical. You can do this by using the property of cube roots that states a33=a\sqrt[3]{a^3} = a.

Q3: What is the difference between a radical and an exponent?

A3: A radical and an exponent are both used to represent repeated multiplication, but they are used in different ways. A radical is used to represent repeated multiplication of a number by itself, while an exponent is used to represent repeated multiplication of a number by itself.

Q4: How do I simplify a radical expression with a variable in the numerator?

A4: To simplify a radical expression with a variable in the numerator, you need to simplify the fraction inside the radical and then simplify the cube root. You can do this by using the property of cube roots that states a33=a\sqrt[3]{a^3} = a.

Q5: What is the final step in simplifying a radical expression?

A5: The final step in simplifying a radical expression is to simplify the fraction. This involves dividing the numerator and denominator by their greatest common divisor (GCD).

Q6: Can I simplify a radical expression with a negative number in the numerator?

A6: Yes, you can simplify a radical expression with a negative number in the numerator. However, you need to be careful when simplifying the fraction inside the radical. You can do this by using the property of cube roots that states a33=a\sqrt[3]{a^3} = a.

Q7: How do I simplify a radical expression with a variable in the denominator?

A7: To simplify a radical expression with a variable in the denominator, you need to simplify the fraction inside the radical and then simplify the cube root. You can do this by using the property of cube roots that states a33=a\sqrt[3]{a^3} = a.

Q8: Can I simplify a radical expression with a fraction in the numerator?

A8: Yes, you can simplify a radical expression with a fraction in the numerator. However, you need to be careful when simplifying the fraction inside the radical. You can do this by using the property of cube roots that states a33=a\sqrt[3]{a^3} = a.

Q9: How do I simplify a radical expression with a variable in the numerator and a variable in the denominator?

A9: To simplify a radical expression with a variable in the numerator and a variable in the denominator, you need to simplify the fraction inside the radical and then simplify the cube root. You can do this by using the property of cube roots that states a33=a\sqrt[3]{a^3} = a.

Q10: Can I simplify a radical expression with a negative number in the denominator?

A10: Yes, you can simplify a radical expression with a negative number in the denominator. However, you need to be careful when simplifying the fraction inside the radical. You can do this by using the property of cube roots that states a33=a\sqrt[3]{a^3} = a.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill to master in mathematics. By following the steps outlined in this article, you should be able to simplify radical expressions with ease. Remember to always simplify the fraction inside the radical and then simplify the cube root. With practice, you will become proficient in simplifying radical expressions.

Practice Problems

Here are some practice problems to help you practice simplifying radical expressions:

  • Simplify x3y12273\sqrt[3]{\frac{x^3 y^{12}}{27}}
  • Simplify x3y121253\sqrt[3]{\frac{x^3 y^{12}}{125}}
  • Simplify x3y122163\sqrt[3]{\frac{x^3 y^{12}}{216}}

Answer Key

Here is the answer key for the practice problems:

  • x3y12273=xy33\sqrt[3]{\frac{x^3 y^{12}}{27}} = \frac{x y^3}{3}
  • x3y121253=xy35\sqrt[3]{\frac{x^3 y^{12}}{125}} = \frac{x y^3}{5}
  • x3y122163=xy36\sqrt[3]{\frac{x^3 y^{12}}{216}} = \frac{x y^3}{6}