Simplify $\sqrt[5]{96 X^5}$.

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Understanding the Problem

When dealing with radicals, it's essential to simplify them by factoring out perfect powers of the index. In this case, we're given the expression $\sqrt[5]{96 x^5}$, and our goal is to simplify it.

Breaking Down the Expression

To simplify the expression, we need to break it down into its prime factors. We can start by factoring the number 96.

Factoring 96

96 can be factored as follows:

96 = 2^5 * 3

So, we can rewrite the expression as:

$\sqrt[5]{2^5 \* 3 \* x^5}$

Simplifying the Radical

Now that we have the prime factors, we can simplify the radical by taking out perfect powers of the index. In this case, the index is 5, so we can take out 5 as a perfect power.

$\sqrt[5]{2^5 \* 3 \* x^5}$ = $\sqrt[5]{(2^5) \* 3 \* (x^5)}$

Simplifying the Perfect Powers

Since 2^5 is a perfect power of 5, we can take it out of the radical.

$\sqrt[5]{(2^5) \* 3 \* (x^5)}$ = $2 \* \sqrt[5]{3 \* x^5}$

Further Simplification

Now that we have simplified the radical, we can further simplify the expression by taking out perfect powers of the index.

Simplifying the Expression

Since x^5 is a perfect power of 5, we can take it out of the radical.

$2 \* \sqrt[5]{3 \* x^5}$ = $2 \* x \* \sqrt[5]{3}$

Conclusion

In conclusion, we have simplified the expression $\sqrt[5]{96 x^5}$ by factoring out perfect powers of the index and taking out perfect powers of the index. The simplified expression is $2 \* x \* \sqrt[5]{3}$.

Final Answer

The final answer is $2 \* x \* \sqrt[5]{3}$.

Related Topics

• Simplifying radicals
• Factoring perfect powers
• Prime factorization

Example Problems

• Simplify $\sqrt[3]{27 x^3}$
• Simplify $\sqrt[4]{16 x^4}$
• Simplify $\sqrt[6]{64 x^6}$

Practice Problems

• Simplify $\sqrt[5]{100 x^5}$
• Simplify $\sqrt[3]{125 x^3}$
• Simplify $\sqrt[4]{256 x^4}$

Additional Resources

• Khan Academy: Simplifying Radicals
• Mathway: Simplifying Radicals
• Wolfram Alpha: Simplifying Radicals
  

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Frequently Asked Questions

Q: What is the index of a radical?

A: The index of a radical is the number outside the radical symbol, which indicates the power to which the radicand is to be raised.

Q: What is the radicand of a radical?

A: The radicand of a radical is the expression inside the radical symbol, which is to be raised to the power indicated by the index.

Q: How do I simplify a radical?

A: To simplify a radical, you need to factor the radicand into its prime factors and take out perfect powers of the index.

Q: What is a perfect power?

A: A perfect power is a number that can be expressed as a power of another number. For example, 4 is a perfect power because it can be expressed as 2^2.

Q: How do I take out perfect powers of the index?

A: To take out perfect powers of the index, you need to identify the perfect powers of the radicand and raise them to the power indicated by the index.

Q: Can I simplify a radical with a variable in the radicand?

A: Yes, you can simplify a radical with a variable in the radicand by following the same steps as you would with a numerical radicand.

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

A: A radical is a mathematical operation that involves taking the nth root of a number, while an exponent is a mathematical operation that involves raising a number to a power.

Q: Can I simplify a radical with a negative exponent?

A: No, you cannot simplify a radical with a negative exponent. However, you can simplify a radical with a negative radicand by multiplying it by the negative sign.

Q: How do I simplify a radical with a coefficient?

A: To simplify a radical with a coefficient, you need to factor the coefficient into its prime factors and take out perfect powers of the index.

Q: Can I simplify a radical with a fraction in the radicand?

A: Yes, you can simplify a radical with a fraction in the radicand by following the same steps as you would with a numerical radicand.

Q: What is the final answer to the problem $\sqrt[5]{96 x^5}$?

A: The final answer to the problem $\sqrt[5]{96 x^5}$ is $2 \* x \* \sqrt[5]{3}$.

Common Mistakes

• Not factoring the radicand into its prime factors
• Not taking out perfect powers of the index
• Not simplifying the radical with a variable in the radicand
• Not simplifying the radical with a negative exponent
• Not simplifying the radical with a coefficient
• Not simplifying the radical with a fraction in the radicand

Tips and Tricks

• Always factor the radicand into its prime factors before simplifying the radical.
• Always take out perfect powers of the index before simplifying the radical.
• Always simplify the radical with a variable in the radicand by following the same steps as you would with a numerical radicand.
• Always simplify the radical with a negative exponent by multiplying it by the negative sign.
• Always simplify the radical with a coefficient by factoring the coefficient into its prime factors and taking out perfect powers of the index.
• Always simplify the radical with a fraction in the radicand by following the same steps as you would with a numerical radicand.

Related Topics

• Simplifying radicals
• Factoring perfect powers
• Prime factorization
• Exponents
• Fractions

Example Problems

• Simplify $\sqrt[3]{27 x^3}$
• Simplify $\sqrt[4]{16 x^4}$
• Simplify $\sqrt[6]{64 x^6}$

Practice Problems

• Simplify $\sqrt[5]{100 x^5}$
• Simplify $\sqrt[3]{125 x^3}$
• Simplify $\sqrt[4]{256 x^4}$

Additional Resources

• Khan Academy: Simplifying Radicals
• Mathway: Simplifying Radicals
• Wolfram Alpha: Simplifying Radicals