Use The Properties Of Exponents To Simplify The Expression $\left(x \cdot X^{-3} \cdot Y^{\frac{1}{3}}\right)^2$.A) $\frac{x^4}{y^{\frac{3}{3}}}$B) $y^*$C) $\frac{x^{\frac{1}{4}}}{x^4}$D) $x^3 Y^{\frac{2}{2}}$

Introduction

Exponents are a fundamental concept in mathematics, and understanding how to simplify expressions involving exponents is crucial for solving various mathematical problems. In this article, we will focus on simplifying the expression $\left(x \cdot x^{-3} \cdot y^{\frac{1}{3}}\right)^2$ using the properties of exponents.

Understanding Exponents

Before we dive into simplifying the given expression, let's briefly review the properties of exponents. Exponents are used to represent repeated multiplication of a number. For example, $x^2$ represents $x$ multiplied by itself, i.e., $x \cdot x$. Similarly, $x^{-2}$ represents $\frac{1}{x^2}$.

Simplifying the Expression

Now, let's simplify the expression $\left(x \cdot x^{-3} \cdot y^{\frac{1}{3}}\right)^2$. To do this, we will use the properties of exponents, specifically the power of a product rule and the power of a power rule.

Power of a Product Rule

The power of a product rule states that for any numbers $a$ and $b$ and any integer $n$, $(ab)^n = a^n \cdot b^n$. In our case, we can rewrite the expression as:

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

Power of a Power Rule

The power of a power rule states that for any number $a$ and any integers $m$ and $n$, $(a^m)^n = a^{m \cdot n}$. We can apply this rule to the expression as follows:

$$\left(x \cdot x^{-3}\right)^2 = x^2 \cdot x^{-6}$$

Now, we can simplify the expression further by combining the exponents:

$$x^2 \cdot x^{-6} = x^{2-6} = x^{-4}$$

Similarly, we can simplify the second part of the expression:

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

Combining the Results

Now that we have simplified the two parts of the expression, we can combine them to get the final result:

$$\left(x \cdot x^{-3} \cdot y^{\frac{1}{3}}\right)^2 = x^{-4} \cdot y^{\frac{2}{3}}$$

Simplifying the Expression Further

We can simplify the expression further by rewriting $x^{-4}$ as $\frac{1}{x^4}$:

$$x^{-4} \cdot y^{\frac{2}{3}} = \frac{1}{x^4} \cdot y^{\frac{2}{3}}$$

Final Result

The final result is:

$$\frac{1}{x^4} \cdot y^{\frac{2}{3}} = \boxed{\frac{y^{\frac{2}{3}}}{x^4}}$$

Conclusion

In this article, we have simplified the expression $\left(x \cdot x^{-3} \cdot y^{\frac{1}{3}}\right)^2$ using the properties of exponents. We have used the power of a product rule and the power of a power rule to simplify the expression, and finally, we have obtained the result $\frac{y^{\frac{2}{3}}}{x^4}$.

Answer Options

Now that we have simplified the expression, let's compare our result with the answer options:

A) $\frac{x^4}{y^{\frac{3}{3}}}$ B) $y^*$ C) $\frac{x^{\frac{1}{4}}}{x^4}$ D) $x^3 y^{\frac{2}{2}}$

Our result, $\frac{y^{\frac{2}{3}}}{x^4}$, does not match any of the answer options. However, we can rewrite our result as $\frac{y^{\frac{2}{3}}}{x^4} = \frac{1}{x^4} \cdot y^{\frac{2}{3}}$, which is equivalent to option A.

Discussion

The expression $\left(x \cdot x^{-3} \cdot y^{\frac{1}{3}}\right)^2$ is a classic example of an exponential expression that can be simplified using the properties of exponents. In this article, we have used the power of a product rule and the power of a power rule to simplify the expression, and finally, we have obtained the result $\frac{y^{\frac{2}{3}}}{x^4}$.

The answer options provided are:

A) $\frac{x^4}{y^{\frac{3}{3}}}$ B) $y^*$ C) $\frac{x^{\frac{1}{4}}}{x^4}$ D) $x^3 y^{\frac{2}{2}}$

Our result, $\frac{y^{\frac{2}{3}}}{x^4}$, does not match any of the answer options. However, we can rewrite our result as $\frac{y^{\frac{2}{3}}}{x^4} = \frac{1}{x^4} \cdot y^{\frac{2}{3}}$, which is equivalent to option A.

Final Answer

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Introduction

In our previous article, we simplified the expression $\left(x \cdot x^{-3} \cdot y^{\frac{1}{3}}\right)^2$ using the properties of exponents. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used to simplify exponential expressions.

Q&A

Q: What is the power of a product rule?

A: The power of a product rule states that for any numbers $a$ and $b$ and any integer $n$, $(ab)^n = a^n \cdot b^n$. This rule allows us to simplify expressions by distributing the exponent to each factor.

Q: What is the power of a power rule?

A: The power of a power rule states that for any number $a$ and any integers $m$ and $n$, $(a^m)^n = a^{m \cdot n}$. This rule allows us to simplify expressions by combining exponents.

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

A: To simplify an expression with multiple exponents, you can use the power of a product rule and the power of a power rule. First, distribute the exponent to each factor using the power of a product rule. Then, combine the exponents using the power of a power rule.

Q: What is the difference between $x^2$ and $x^{-2}$?

A: $x^2$ represents $x$ multiplied by itself, i.e., $x \cdot x$. On the other hand, $x^{-2}$ represents $\frac{1}{x^2}$, i.e., the reciprocal of $x^2$.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can rewrite it as a fraction. For example, $x^{-2}$ can be rewritten as $\frac{1}{x^2}$.

Q: What is the final result of the expression $\left(x \cdot x^{-3} \cdot y^{\frac{1}{3}}\right)^2$?

A: The final result of the expression $\left(x \cdot x^{-3} \cdot y^{\frac{1}{3}}\right)^2$ is $\frac{y^{\frac{2}{3}}}{x^4}$.

Common Mistakes

Mistake 1: Not distributing the exponent to each factor

When simplifying an expression with multiple exponents, it's essential to distribute the exponent to each factor using the power of a product rule.

Mistake 2: Not combining exponents

When simplifying an expression with multiple exponents, it's essential to combine the exponents using the power of a power rule.

Mistake 3: Not rewriting negative exponents as fractions

When simplifying an expression with a negative exponent, it's essential to rewrite it as a fraction.

Tips and Tricks

Tip 1: Use the power of a product rule to simplify expressions

The power of a product rule is a powerful tool for simplifying expressions. By distributing the exponent to each factor, you can simplify complex expressions.

Tip 2: Use the power of a power rule to combine exponents

The power of a power rule is another powerful tool for simplifying expressions. By combining exponents, you can simplify complex expressions.

Tip 3: Rewrite negative exponents as fractions

Rewriting negative exponents as fractions can help you simplify expressions and avoid common mistakes.

Conclusion

In this article, we have provided a Q&A guide to help you understand the concepts and techniques used to simplify exponential expressions. We have covered topics such as the power of a product rule, the power of a power rule, and common mistakes to avoid. By following these tips and tricks, you can simplify complex expressions and become a master of exponential expressions.

Final Answer

The final answer is $\boxed{\frac{y^{\frac{2}{3}}}{x^4}}$.