Which Expression Is Equivalent To $\left(x^{27} Y\right)^{\frac{1}{3}}$?A. $x^3(\sqrt[3]{y}$\] B. $x^9(\sqrt[3]{y}$\] C. $x^{27}(\sqrt[3]{y}$\] D. $x^{24}(\sqrt[3]{y}$\]

Introduction

In mathematics, simplifying exponential expressions is a crucial skill that helps us to manipulate and solve equations more efficiently. One of the key concepts in simplifying exponential expressions is understanding the properties of exponents, particularly the power of a power property. In this article, we will explore how to simplify the expression $\left(x^{27} y\right)^{\frac{1}{3}}$ and find its equivalent form.

Understanding the Power of a Power Property

The power of a power property states that for any numbers $a$, $b$, and $c$, we have:

$$\left(a^b\right)^c = a^{bc}$$

This property allows us to simplify expressions by combining the exponents. In the given expression $\left(x^{27} y\right)^{\frac{1}{3}}$, we can apply the power of a power property to simplify it.

Simplifying the Expression

Using the power of a power property, we can rewrite the expression as:

$$\left(x^{27} y\right)^{\frac{1}{3}} = x^{\frac{1}{3} \cdot 27} y^{\frac{1}{3}}$$

Simplifying the exponent, we get:

$$x^{\frac{1}{3} \cdot 27} y^{\frac{1}{3}} = x^9 y^{\frac{1}{3}}$$

Now, we can rewrite $y^{\frac{1}{3}}$ as $\sqrt[3]{y}$, which is the cube root of $y$. Therefore, the simplified expression is:

$$x^9 \sqrt[3]{y}$$

Comparing with the Options

Now that we have simplified the expression, let's compare it with the given options:

A. $x^3(\sqrt[3]{y}$] B. $x^9(\sqrt[3]{y}$] C. $x^{27}(\sqrt[3]{y}$] D. $x^{24}(\sqrt[3]{y}$]

From our simplified expression, we can see that the correct answer is:

B. $x^9(\sqrt[3]{y}$]

Conclusion

In this article, we have explored how to simplify the expression $\left(x^{27} y\right)^{\frac{1}{3}}$ using the power of a power property. We have shown that the simplified expression is equivalent to $x^9 \sqrt[3]{y}$. By comparing this with the given options, we have found that the correct answer is option B. This exercise demonstrates the importance of understanding the properties of exponents and how to apply them to simplify complex expressions.

Additional Tips and Examples

Here are some additional tips and examples to help you practice simplifying exponential expressions:

• Tip 1: When simplifying exponential expressions, always start by applying the power of a power property.
• Tip 2: Make sure to simplify the exponents by multiplying or dividing them as needed.
• Example 1: Simplify the expression $\left(x^4 y\right)^2$ using the power of a power property.
• Example 2: Simplify the expression $\left(x^3 y\right)^{\frac{1}{2}}$ using the power of a power property.

By following these tips and practicing with examples, you will become more confident in simplifying exponential expressions and solving equations more efficiently.

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying exponential expressions:

• Mistake 1: Failing to apply the power of a power property when simplifying expressions.
• Mistake 2: Not simplifying the exponents by multiplying or dividing them as needed.
• Mistake 3: Not rewriting the expression in a simplified form.

By being aware of these common mistakes, you can avoid them and ensure that your simplifications are accurate and efficient.

Conclusion

Q&A: Simplifying Exponential Expressions

Q: What is the power of a power property?

A: The power of a power property states that for any numbers $a$, $b$, and $c$, we have:

$$\left(a^b\right)^c = a^{bc}$$

This property allows us to simplify expressions by combining the exponents.

Q: How do I simplify the expression $\left(x^{27} y\right)^{\frac{1}{3}}$?

A: To simplify the expression, we can apply the power of a power property:

$$\left(x^{27} y\right)^{\frac{1}{3}} = x^{\frac{1}{3} \cdot 27} y^{\frac{1}{3}}$$

Simplifying the exponent, we get:

$$x^{\frac{1}{3} \cdot 27} y^{\frac{1}{3}} = x^9 y^{\frac{1}{3}}$$

Now, we can rewrite $y^{\frac{1}{3}}$ as $\sqrt[3]{y}$, which is the cube root of $y$. Therefore, the simplified expression is:

$$x^9 \sqrt[3]{y}$$

Q: What is the difference between $x^9$ and $x^{27}$?

A: The difference between $x^9$ and $x^{27}$ is the exponent. In $x^9$, the exponent is 9, while in $x^{27}$, the exponent is 27. When we simplify the expression $\left(x^{27} y\right)^{\frac{1}{3}}$, we get $x^9 \sqrt[3]{y}$, which means that the exponent is reduced from 27 to 9.

Q: Can I simplify the expression $\left(x^4 y\right)^2$ using the power of a power property?

A: Yes, you can simplify the expression $\left(x^4 y\right)^2$ using the power of a power property:

$$\left(x^4 y\right)^2 = x^{4 \cdot 2} y^2$$

Simplifying the exponent, we get:

$$x^{4 \cdot 2} y^2 = x^8 y^2$$

Q: What is the cube root of a number?

A: The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3, because $3 \cdot 3 \cdot 3 = 27$.

Q: Can I simplify the expression $\left(x^3 y\right)^{\frac{1}{2}}$ using the power of a power property?

A: Yes, you can simplify the expression $\left(x^3 y\right)^{\frac{1}{2}}$ using the power of a power property:

$$\left(x^3 y\right)^{\frac{1}{2}} = x^{\frac{1}{2} \cdot 3} y^{\frac{1}{2}}$$

Simplifying the exponent, we get:

$$x^{\frac{1}{2} \cdot 3} y^{\frac{1}{2}} = x^{\frac{3}{2}} y^{\frac{1}{2}}$$

Now, we can rewrite $y^{\frac{1}{2}}$ as $\sqrt{y}$, which is the square root of $y$. Therefore, the simplified expression is:

$$x^{\frac{3}{2}} \sqrt{y}$$

Conclusion

In this Q&A article, we have explored some common questions and answers related to simplifying exponential expressions. We have discussed the power of a power property, how to simplify expressions using this property, and some examples of simplifying expressions. By following the tips and examples provided in this article, you will become more confident in simplifying exponential expressions and solving equations more efficiently.