Simplify $\sqrt[3]{16 X^7 Y^{14}}$.

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Understanding the Problem

When dealing with radicals, it's essential to understand the properties of exponents and roots. In this problem, we're given the expression $\sqrt[3]{16 x^7 y^{14}}$, and we're asked to simplify it. To simplify this expression, we need to use the properties of radicals and exponents.

Breaking Down the Expression

The given expression can be broken down into three parts: the coefficient, the variable $x$, and the variable $y$. The coefficient is $16$, which can be written as $2^4$. The variable $x$ has an exponent of $7$, and the variable $y$ has an exponent of $14$.

Using the Properties of Radicals

To simplify the expression, we can use the property of radicals that states $\sqrt[n]{a^n} = a$. This property allows us to simplify the expression by taking out the cube root of the coefficient, the variable $x$, and the variable $y$.

Simplifying the Coefficient

The coefficient $16$ can be written as $2^4$. Taking the cube root of $2^4$ gives us $2^{4/3}$. Since $4/3$ is not a whole number, we cannot simplify it further.

Simplifying the Variable $x$

The variable $x$ has an exponent of $7$. Taking the cube root of $x^7$ gives us $x^{7/3}$.

Simplifying the Variable $y$

The variable $y$ has an exponent of $14$. Taking the cube root of $y^{14}$ gives us $y^{14/3}$.

Combining the Simplified Parts

Now that we have simplified the coefficient, the variable $x$, and the variable $y$, we can combine them to get the final simplified expression.

The Final Simplified Expression

The final simplified expression is $\boxed{2^{4/3} x^{7/3} y^{14/3}}$.

Conclusion

In this problem, we used the properties of radicals and exponents to simplify the expression $\sqrt[3]{16 x^7 y^{14}}$. We broke down the expression into three parts: the coefficient, the variable $x$, and the variable $y$. We then used the property of radicals to simplify each part and combined them to get the final simplified expression.

Additional Tips and Tricks

When dealing with radicals, it's essential to remember the properties of exponents and roots. Here are some additional tips and tricks to help you simplify radicals:

• Use the property of radicals that states $\sqrt[n]{a^n} = a$ to simplify the expression.
• Break down the expression into three parts: the coefficient, the variable, and the exponent.
• Use the property of radicals to simplify each part.
• Combine the simplified parts to get the final simplified expression.

Common Mistakes to Avoid

When dealing with radicals, it's essential to avoid common mistakes. Here are some common mistakes to avoid:

• Not using the property of radicals to simplify the expression.
• Not breaking down the expression into three parts: the coefficient, the variable, and the exponent.
• Not using the property of radicals to simplify each part.
• Not combining the simplified parts to get the final simplified expression.

Real-World Applications

Radicals have many real-world applications. Here are some examples:

• In physics, radicals are used to describe the motion of objects.
• In engineering, radicals are used to describe the stress on materials.
• In finance, radicals are used to describe the growth of investments.

Final Thoughts

In conclusion, simplifying radicals is an essential skill in mathematics. By understanding the properties of exponents and roots, we can simplify radicals and solve complex problems. Remember to use the property of radicals to simplify the expression, break down the expression into three parts, and combine the simplified parts to get the final simplified expression. With practice and patience, you can become proficient in simplifying radicals and solving complex problems.

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Frequently Asked Questions

Q: What is the property of radicals that we use to simplify the expression?

A: The property of radicals that we use to simplify the expression is $\sqrt[n]{a^n} = a$. This property allows us to simplify the expression by taking out the cube root of the coefficient, the variable $x$, and the variable $y$.

Q: How do we simplify the coefficient in the expression?

A: To simplify the coefficient, we can write it as $2^4$. Taking the cube root of $2^4$ gives us $2^{4/3}$. Since $4/3$ is not a whole number, we cannot simplify it further.

Q: How do we simplify the variable $x$ in the expression?

A: To simplify the variable $x$, we can take the cube root of $x^7$, which gives us $x^{7/3}$.

Q: How do we simplify the variable $y$ in the expression?

A: To simplify the variable $y$, we can take the cube root of $y^{14}$, which gives us $y^{14/3}$.

Q: How do we combine the simplified parts to get the final simplified expression?

A: To combine the simplified parts, we can multiply the simplified coefficient, the simplified variable $x$, and the simplified variable $y$ together. This gives us the final simplified expression: $2^{4/3} x^{7/3} y^{14/3}$.

Q: What are some common mistakes to avoid when simplifying radicals?

A: Some common mistakes to avoid when simplifying radicals include not using the property of radicals to simplify the expression, not breaking down the expression into three parts: the coefficient, the variable, and the exponent, and not combining the simplified parts to get the final simplified expression.

Q: What are some real-world applications of radicals?

A: Radicals have many real-world applications, including physics, engineering, and finance. In physics, radicals are used to describe the motion of objects. In engineering, radicals are used to describe the stress on materials. In finance, radicals are used to describe the growth of investments.

Q: How can I practice simplifying radicals?

A: To practice simplifying radicals, you can try simplifying different expressions using the property of radicals. You can also try breaking down expressions into three parts: the coefficient, the variable, and the exponent, and then combining the simplified parts to get the final simplified expression.

Q: What are some tips for simplifying radicals?

A: Some tips for simplifying radicals include using the property of radicals to simplify the expression, breaking down the expression into three parts: the coefficient, the variable, and the exponent, and combining the simplified parts to get the final simplified expression. You can also try to simplify the expression by taking out the cube root of the coefficient, the variable $x$, and the variable $y$.

Additional Resources

• For more information on radicals, see the article on Radicals.
• For more practice problems on simplifying radicals, see the article on Simplifying Radicals.
• For more information on real-world applications of radicals, see the article on Real-World Applications of Radicals.

Conclusion

Simplifying radicals is an essential skill in mathematics. By understanding the properties of exponents and roots, we can simplify radicals and solve complex problems. Remember to use the property of radicals to simplify the expression, break down the expression into three parts, and combine the simplified parts to get the final simplified expression. With practice and patience, you can become proficient in simplifying radicals and solving complex problems.