Simplify: $\sqrt[4]{5^4}$A. 4 B. 5 C. 25 D. 625

Understanding the Problem

The given problem involves simplifying a radical expression, specifically a fourth root. The expression $\sqrt[4]{5^4}$ requires us to find the fourth root of $5^4$. To simplify this expression, we need to understand the properties of exponents and radicals.

Properties of Exponents and Radicals

When dealing with exponents and radicals, it's essential to remember the following properties:

• Exponent Rule: When a power is raised to another power, the exponents are multiplied. For example, $(a^m)^n = a^{m \cdot n}$.
• Radical Rule: The nth root of a number can be expressed as a fractional exponent. For example, $\sqrt[n]{a} = a^{\frac{1}{n}}$.

Simplifying the Expression

Using the properties of exponents and radicals, we can simplify the expression $\sqrt[4]{5^4}$ as follows:

$\sqrt[4]{5^4} = (5^4)^{\frac{1}{4}}$

Applying the exponent rule, we get:

$(5^4)^{\frac{1}{4}} = 5^{4 \cdot \frac{1}{4}}$

Simplifying the exponent, we get:

$5^{4 \cdot \frac{1}{4}} = 5^1$

Therefore, the simplified expression is:

$\sqrt[4]{5^4} = 5$

Conclusion

In conclusion, the correct answer to the problem is $\boxed{5}$. This is because the fourth root of $5^4$ is simply $5$, as the exponent rule and radical rule allow us to simplify the expression.

Final Answer

The final answer is $\boxed{5}$.

Related Problems

If you're interested in practicing more problems like this, here are a few related problems:

• Simplify: $\sqrt[3]{8^3}$
• Simplify: $\sqrt[2]{9^2}$
• Simplify: $\sqrt[5]{2^5}$

These problems involve simplifying radical expressions using the properties of exponents and radicals.

Frequently Asked Questions

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

A: A radical is a mathematical operation that involves finding the nth root of a number, while an exponent is a power to which a number is raised. In the expression $\sqrt[4]{5^4}$, the 4th root is a radical, and the 4 is an exponent.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to understand the properties of exponents and radicals. You can use the exponent rule to simplify the expression by multiplying the exponents, and then simplify the resulting expression.

Q: What is the property of exponents that allows us to simplify the expression $\sqrt[4]{5^4}$?

A: The property of exponents that allows us to simplify the expression $\sqrt[4]{5^4}$ is the exponent rule, which states that when a power is raised to another power, the exponents are multiplied.

Q: How do I apply the exponent rule to simplify the expression $\sqrt[4]{5^4}$?

A: To apply the exponent rule, you need to multiply the exponents. In the expression $\sqrt[4]{5^4}$, the exponent is 4, and the power is also 4. Multiplying the exponents, you get $4 \cdot \frac{1}{4} = 1$. Therefore, the simplified expression is $5^1 = 5$.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{5}$.

Q: Can I use a calculator to simplify the expression $\sqrt[4]{5^4}$?

A: While a calculator can be useful for simplifying expressions, it's not necessary in this case. The expression $\sqrt[4]{5^4}$ can be simplified using the properties of exponents and radicals, as shown above.

Q: How do I practice simplifying radical expressions?

A: To practice simplifying radical expressions, you can try solving problems like the one above. You can also try simplifying different types of radical expressions, such as $\sqrt[3]{8^3}$ or $\sqrt[2]{9^2}$.

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

A: Some common mistakes to avoid when simplifying radical expressions include:

• Not understanding the properties of exponents and radicals
• Not applying the exponent rule correctly
• Not simplifying the resulting expression
• Not checking the final answer for accuracy

Additional Resources

If you're interested in learning more about simplifying radical expressions, here are some additional resources:

• Khan Academy: Simplifying Radical Expressions
• Mathway: Simplifying Radical Expressions
• Wolfram Alpha: Simplifying Radical Expressions

These resources provide additional practice problems, explanations, and examples to help you understand and simplify radical expressions.

Conclusion

In conclusion, simplifying radical expressions requires understanding the properties of exponents and radicals, as well as applying the exponent rule correctly. By following the steps outlined above, you can simplify radical expressions like $\sqrt[4]{5^4}$ and arrive at the final answer of $\boxed{5}$.