Simplify: $\left(25 P^4\right)^{\frac{1}{2}}$A. $p^2$ B. $8 P$ C. $5 P^2$ D. $5 P^8$

Understanding Exponents and Radicals

Exponents and radicals are fundamental concepts in mathematics that help us simplify complex expressions. In this article, we will focus on simplifying expressions involving exponents and radicals. Specifically, we will tackle the expression $\left(25 p^4\right)^{\frac{1}{2}}$ and explore the different options provided.

The Expression: $\left(25 p^4\right)^{\frac{1}{2}}$

The given expression involves a radical, which is a mathematical operation that extracts the square root of a number. In this case, we have $\left(25 p^4\right)^{\frac{1}{2}}$, which can be read as "the square root of $25 p^4$".

Simplifying the Expression

To simplify the expression, we need to apply the rules of exponents and radicals. The first step is to recognize that the expression can be rewritten as $\sqrt{25 p^4}$. This is because the square root of a product is equal to the product of the square roots.

Applying the Rules of Exponents and Radicals

Now, let's apply the rules of exponents and radicals to simplify the expression. We know that the square root of a product is equal to the product of the square roots. Therefore, we can rewrite the expression as $\sqrt{25} \cdot \sqrt{p^4}$.

Simplifying the Square Roots

The square root of 25 is 5, since $5^2 = 25$. Therefore, we can simplify the expression as $5 \cdot \sqrt{p^4}$.

Applying the Power Rule

The next step is to apply the power rule, which states that $\sqrt{x^n} = x^{\frac{n}{2}}$. In this case, we have $\sqrt{p^4}$, which can be rewritten as $p^{\frac{4}{2}}$.

Simplifying the Expression Further

Now, let's simplify the expression further. We know that $p^{\frac{4}{2}} = p^2$. Therefore, we can rewrite the expression as $5 \cdot p^2$.

Evaluating the Options

Now that we have simplified the expression, let's evaluate the options provided.

• Option A: $p^2$
• Option B: $8 p$
• Option C: $5 p^2$
• Option D: $5 p^8$

Conclusion

Based on our simplification, we can see that the correct answer is Option C: $5 p^2$. This is because we simplified the expression to $5 \cdot p^2$, which matches the format of Option C.

Final Answer

Q&A: Simplifying Exponents and Radicals

In the previous article, we explored the concept of simplifying exponents and radicals. We tackled the expression $\left(25 p^4\right)^{\frac{1}{2}}$ and simplified it to $5 p^2$. In this article, we will provide a Q&A section to help you better understand the concepts and apply them to different scenarios.

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

A: An exponent is a mathematical operation that represents repeated multiplication of a number. For example, $2^3$ means $2 \times 2 \times 2$. A radical, on the other hand, is a mathematical operation that extracts the square root of a number. For example, $\sqrt{4}$ means the number that, when multiplied by itself, gives 4.

Q: How do I simplify an expression with a radical?

A: To simplify an expression with a radical, you need to apply the rules of exponents and radicals. The first step is to recognize that the expression can be rewritten as the product of the square roots of its factors. Then, you can simplify each factor separately and multiply the results.

Q: What is the power rule for radicals?

A: The power rule for radicals states that $\sqrt{x^n} = x^{\frac{n}{2}}$. This means that you can rewrite a radical expression as a power of the number inside the radical.

Q: How do I apply the power rule for radicals?

A: To apply the power rule for radicals, you need to identify the exponent of the number inside the radical. Then, you can rewrite the radical expression as a power of the number, using the exponent as the new exponent.

Q: What is the difference between $\sqrt{x}$ and $x^{\frac{1}{2}}$?

A: $\sqrt{x}$ and $x^{\frac{1}{2}}$ are equivalent expressions. The square root of a number is the number that, when multiplied by itself, gives the original number. The power of $\frac{1}{2}$ represents the same operation.

Q: How do I simplify an expression with multiple radicals?

A: To simplify an expression with multiple radicals, you need to apply the rules of exponents and radicals. The first step is to recognize that the expression can be rewritten as the product of the square roots of its factors. Then, you can simplify each factor separately and multiply the results.

Q: What is the rule for multiplying radicals?

A: The rule for multiplying radicals states that $\sqrt{x} \cdot \sqrt{y} = \sqrt{x \cdot y}$. This means that you can multiply two radical expressions by multiplying the numbers inside the radicals.

Q: How do I apply the rule for multiplying radicals?

A: To apply the rule for multiplying radicals, you need to multiply the numbers inside the radicals. Then, you can rewrite the result as a single radical expression.

Q: What is the rule for dividing radicals?

A: The rule for dividing radicals states that $\frac{\sqrt{x}}{\sqrt{y}} = \sqrt{\frac{x}{y}}$. This means that you can divide two radical expressions by dividing the numbers inside the radicals.

Q: How do I apply the rule for dividing radicals?

A: To apply the rule for dividing radicals, you need to divide the numbers inside the radicals. Then, you can rewrite the result as a single radical expression.

Conclusion

In this Q&A article, we provided answers to common questions about simplifying exponents and radicals. We hope that this article has helped you better understand the concepts and apply them to different scenarios. Remember to always apply the rules of exponents and radicals when simplifying expressions with radicals.

Final Tips

• Always apply the rules of exponents and radicals when simplifying expressions with radicals.
• Recognize that the expression can be rewritten as the product of the square roots of its factors.
• Simplify each factor separately and multiply the results.
• Apply the power rule for radicals to rewrite a radical expression as a power of the number inside the radical.
• Multiply and divide radical expressions by multiplying and dividing the numbers inside the radicals.