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 $a$ and $b$ and any integers $m$ and $n$, we have:

$$\left(a^m\right)^n = a^{m \cdot n}$$

This property allows us to simplify expressions by multiplying the exponents when we have a power raised to a power.

Simplifying the Given Expression

Now, let's apply the power of a power property to simplify the given expression $\left(x^{27} y\right)^{\frac{1}{3}}$. We can rewrite the expression as:

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

Using the power of a power property, we can simplify the first term as:

$$\left(x^{27}\right)^{\frac{1}{3}} = x^{27 \cdot \frac{1}{3}} = x^9$$

So, the expression becomes:

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

Finding the Equivalent Form

Now, we need to find the equivalent form of the expression $x^9 \cdot y^{\frac{1}{3}}$. We can rewrite the expression as:

$$x^9 \cdot y^{\frac{1}{3}} = x^9 \cdot \sqrt[3]{y}$$

This is the equivalent form of the given expression.

Comparing with the Options

Let's compare the equivalent form $x^9 \cdot \sqrt[3]{y}$ with the options provided:

• A. $x^3(\sqrt[3]{y}$: This option is incorrect because the exponent of $x$ is not $9$.
• B. $x^9(\sqrt[3]{y}$: This option is correct because it matches the equivalent form we derived.
• C. $x^{27}(\sqrt[3]{y}$: This option is incorrect because the exponent of $x$ is not $9$.
• D. $x^{24}(\sqrt[3]{y}$: This option is incorrect because the exponent of $x$ is not $9$.

Conclusion

In this article, we simplified the expression $\left(x^{27} y\right)^{\frac{1}{3}}$ using the power of a power property and found its equivalent form. We compared the equivalent form with the options provided and identified the correct answer. This exercise demonstrates the importance of understanding the properties of exponents and how to apply them to simplify complex expressions.

Key Takeaways

• The power of a power property states that for any numbers $a$ and $b$ and any integers $m$ and $n$, we have $\left(a^m\right)^n = a^{m \cdot n}$.
• We can simplify the expression $\left(x^{27} y\right)^{\frac{1}{3}}$ by applying the power of a power property.
• The equivalent form of the expression is $x^9 \cdot \sqrt[3]{y}$.
• We can compare the equivalent form with the options provided to identify the correct answer.

Final Answer

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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 $a$ and $b$ and any integers $m$ and $n$, we have:

$$\left(a^m\right)^n = a^{m \cdot n}$$

This property allows us to simplify expressions by multiplying the exponents when we have a power raised to a power.

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

A: To simplify the expression $\left(x^{27} y\right)^{\frac{1}{3}}$, we can apply the power of a power property. We can rewrite the expression as:

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

Using the power of a power property, we can simplify the first term as:

$$\left(x^{27}\right)^{\frac{1}{3}} = x^{27 \cdot \frac{1}{3}} = x^9$$

So, the expression becomes:

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

Q: What is the equivalent form of the expression $x^9 \cdot y^{\frac{1}{3}}$?

A: The equivalent form of the expression $x^9 \cdot y^{\frac{1}{3}}$ is:

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

This is the simplified form of the original expression.

Q: How do I compare the equivalent form with the options provided?

A: To compare the equivalent form with the options provided, we can simply look at the exponents and the radical expressions. In this case, the equivalent form is $x^9 \cdot \sqrt[3]{y}$, which matches option B.

Q: What is the final answer?

A: The final answer is B. $x^9(\sqrt[3]{y}$.

Common Mistakes to Avoid

• Not applying the power of a power property when simplifying expressions.
• Not multiplying the exponents when applying the power of a power property.
• Not comparing the equivalent form with the options provided.

Tips and Tricks

• Make sure to apply the power of a power property when simplifying expressions.
• Multiply the exponents when applying the power of a power property.
• Compare the equivalent form with the options provided to ensure accuracy.

Conclusion

In this article, we provided a guide to simplifying exponential expressions using the power of a power property. We also answered common questions and provided tips and tricks to help you avoid common mistakes. By following these guidelines, you can simplify complex expressions and find their equivalent forms with ease.