Simplify The Expression: $\[ \frac{\left(x^{-3}\right)^4 X^4}{2 X^{-3}} \\]

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of the rules of exponents. In this article, we will focus on simplifying the given expression: $\frac{\left(x^{-3}\right)^4 x^4}{2 x^{-3}}$. We will use the rules of exponents to simplify the expression and provide a step-by-step solution.

Understanding the Rules of Exponents

Before we dive into simplifying the expression, let's review the rules of exponents. The rules of exponents state that:

• When we raise a power to a power, we multiply the exponents: $(a^m)^n = a^{m \cdot n}$
• When we multiply powers with the same base, we add the exponents: $a^m \cdot a^n = a^{m + n}$
• When we divide powers with the same base, we subtract the exponents: $\frac{a^m}{a^n} = a^{m - n}$
• When we have a power with a negative exponent, we can rewrite it as a fraction: $a^{-m} = \frac{1}{a^m}$

Simplifying the Expression

Now that we have reviewed the rules of exponents, let's simplify the given expression: $\frac{\left(x^{-3}\right)^4 x^4}{2 x^{-3}}$.

Step 1: Simplify the numerator

Using the rule of exponents that states $(a^m)^n = a^{m \cdot n}$, we can simplify the numerator as follows:

$\left(x^{-3}\right)^4 = x^{-3 \cdot 4} = x^{-12}$

So, the numerator becomes: $x^{-12} x^4$

Step 2: Simplify the numerator using the rule of exponents

Using the rule of exponents that states $a^m \cdot a^n = a^{m + n}$, we can simplify the numerator as follows:

$x^{-12} x^4 = x^{-12 + 4} = x^{-8}$

Step 3: Simplify the denominator

Using the rule of exponents that states $\frac{a^m}{a^n} = a^{m - n}$, we can simplify the denominator as follows:

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

Step 4: Simplify the expression

Now that we have simplified the numerator and denominator, we can simplify the expression as follows:

$\frac{x^{-8}}{\frac{1}{2} x^{3}} = \frac{x^{-8}}{x^3} \cdot \frac{2}{1}$

Using the rule of exponents that states $\frac{a^m}{a^n} = a^{m - n}$, we can simplify the expression as follows:

$\frac{x^{-8}}{x^3} = x^{-8 - 3} = x^{-11}$

So, the expression becomes: $x^{-11} \cdot 2$

Step 5: Simplify the expression using the rule of exponents

Using the rule of exponents that states $a^m \cdot a^n = a^{m + n}$, we can simplify the expression as follows:

$x^{-11} \cdot 2 = 2 x^{-11}$

Conclusion

In this article, we simplified the expression: $\frac{\left(x^{-3}\right)^4 x^4}{2 x^{-3}}$ using the rules of exponents. We reviewed the rules of exponents and applied them to simplify the expression step-by-step. The final simplified expression is: $2 x^{-11}$.

Final Answer

The final answer is: $\boxed{2 x^{-11}}$

Discussion

The expression $\frac{\left(x^{-3}\right)^4 x^4}{2 x^{-3}}$ is a complex expression that requires a deep understanding of the rules of exponents. By simplifying the expression step-by-step, we can arrive at the final simplified expression: $2 x^{-11}$. This expression can be used in various mathematical applications, such as solving equations and inequalities.

Related Topics

• Simplifying algebraic expressions
• Rules of exponents
• Negative exponents
• Fractional exponents

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Precalculus" by Michael Sullivan

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.

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Introduction

In our previous article, we simplified the expression: $\frac{\left(x^{-3}\right)^4 x^4}{2 x^{-3}}$ using the rules of exponents. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q&A

Q: What is the rule of exponents that states $(a^m)^n = a^{m \cdot n}$?

A: The rule of exponents that states $(a^m)^n = a^{m \cdot n}$ is a fundamental rule in algebra that allows us to simplify expressions with exponents. When we raise a power to a power, we multiply the exponents.

Q: How do we simplify the expression $\frac{x^{-8}}{x^3}$?

A: To simplify the expression $\frac{x^{-8}}{x^3}$, we use the rule of exponents that states $\frac{a^m}{a^n} = a^{m - n}$. In this case, we subtract the exponents: $x^{-8 - 3} = x^{-11}$.

Q: What is the final simplified expression?

A: The final simplified expression is: $2 x^{-11}$.

Q: Can we simplify the expression further?

A: No, the expression $2 x^{-11}$ is already simplified. We have applied all the rules of exponents and cannot simplify it further.

Q: What is the significance of the negative exponent $x^{-11}$?

A: The negative exponent $x^{-11}$ indicates that we are dividing 1 by $x^{11}$. In other words, $x^{-11} = \frac{1}{x^{11}}$.

Q: Can we rewrite the expression $2 x^{-11}$ as a fraction?

A: Yes, we can rewrite the expression $2 x^{-11}$ as a fraction: $\frac{2}{x^{11}}$.

Q: What is the relationship between the expression $2 x^{-11}$ and the original expression $\frac{\left(x^{-3}\right)^4 x^4}{2 x^{-3}}$?

A: The expression $2 x^{-11}$ is the simplified form of the original expression $\frac{\left(x^{-3}\right)^4 x^4}{2 x^{-3}}$. We applied the rules of exponents to simplify the original expression and arrived at the final simplified expression.

Conclusion

In this Q&A article, we have addressed some common questions and doubts that readers may have regarding the simplification of the expression: $\frac{\left(x^{-3}\right)^4 x^4}{2 x^{-3}}$. We have provided step-by-step explanations and examples to help clarify any confusion.

Final Answer

The final answer is: $\boxed{2 x^{-11}}$

Discussion

The expression $\frac{\left(x^{-3}\right)^4 x^4}{2 x^{-3}}$ is a complex expression that requires a deep understanding of the rules of exponents. By simplifying the expression step-by-step, we can arrive at the final simplified expression: $2 x^{-11}$. This expression can be used in various mathematical applications, such as solving equations and inequalities.

Related Topics

• Simplifying algebraic expressions
• Rules of exponents
• Negative exponents
• Fractional exponents

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Precalculus" by Michael Sullivan

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.