What Is The Simplest Form Of $(5x)^{-3}\left(25x^{-8}\right)$ In Terms Of A Positive Exponent?A. $\frac{25}{x^{11}}$B. \$\frac{1}{5x^{11}}$[/tex\]C. $\frac{x^8}{3}$D. $\frac{1}{3x^{24}}$

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Understanding Exponents and Their Rules

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In this article, we will explore the simplest form of the expression $(5x){-3}\left(25x{-8}\right)$ in terms of a positive exponent.

The Rules of Exponents

Before we dive into the simplification process, it's essential to understand the rules of exponents. The rules are as follows:

  • Product Rule: When multiplying two numbers with the same base, add their exponents. For example, $a^m \cdot a^n = a^{m+n}$
  • Power Rule: When raising a power to another power, multiply the exponents. For example, $(am)n = a^{m \cdot n}$
  • Quotient Rule: When dividing two numbers with the same base, subtract their exponents. For example, $\frac{am}{an} = a^{m-n}$
  • Zero Exponent Rule: Any number raised to the power of zero is equal to 1. For example, $a^0 = 1$

Simplifying the Expression

Now that we have a solid understanding of the rules of exponents, let's simplify the expression $(5x){-3}\left(25x{-8}\right)$.

First, we can rewrite the expression as $\frac{(5x)^{-3}}{1} \cdot \frac{25x^{-8}}{1}$.

Next, we can apply the quotient rule to simplify the expression. We have $\frac{(5x)^{-3}}{1} \cdot \frac{25x^{-8}}{1} = \frac{(5x)^{-3} \cdot 25x^{-8}}{1}$.

Now, we can apply the product rule to simplify the expression. We have $\frac{(5x)^{-3} \cdot 25x^{-8}}{1} = \frac{5^{-3} \cdot 25 \cdot x^{-3} \cdot x^{-8}}{1}$.

Next, we can apply the power rule to simplify the expression. We have $\frac{5^{-3} \cdot 25 \cdot x^{-3} \cdot x^{-8}}{1} = \frac{5^{-3} \cdot 5^2 \cdot x^{-3} \cdot x^{-8}}{1}$.

Now, we can apply the product rule to simplify the expression. We have $\frac{5^{-3} \cdot 5^2 \cdot x^{-3} \cdot x^{-8}}{1} = \frac{5^{-3+2} \cdot x^{-3-8}}{1}$.

Next, we can apply the power rule to simplify the expression. We have $\frac{5^{-3+2} \cdot x^{-3-8}}{1} = \frac{5^{-1} \cdot x^{-11}}{1}$.

Finally, we can rewrite the expression in a more simplified form. We have $\frac{5^{-1} \cdot x^{-11}}{1} = \frac{1}{5} \cdot \frac{1}{x^{11}}$.

The Final Answer

The simplest form of the expression $(5x){-3}\left(25x{-8}\right)$ in terms of a positive exponent is $\frac{1}{5x^{11}}$.

Conclusion

In this article, we have explored the simplest form of the expression $(5x){-3}\left(25x{-8}\right)$ in terms of a positive exponent. We have applied the rules of exponents, including the product rule, power rule, quotient rule, and zero exponent rule, to simplify the expression. The final answer is $\frac{1}{5x^{11}}$.

Answer Key

The correct answer is B. $\frac{1}{5x^{11}}$.

Additional Resources

For more information on exponents and their rules, please refer to the following resources:

  • Khan Academy: Exponents and Exponential Functions
  • Mathway: Exponents and Exponential Functions
  • Wolfram Alpha: Exponents and Exponential Functions

Final Thoughts

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about simplifying exponents.

Q: What is the rule for simplifying exponents?

A: The rule for simplifying exponents is as follows:

  • Product Rule: When multiplying two numbers with the same base, add their exponents. For example, $a^m \cdot a^n = a^{m+n}$
  • Power Rule: When raising a power to another power, multiply the exponents. For example, $(am)n = a^{m \cdot n}$
  • Quotient Rule: When dividing two numbers with the same base, subtract their exponents. For example, $\frac{am}{an} = a^{m-n}$
  • Zero Exponent Rule: Any number raised to the power of zero is equal to 1. For example, $a^0 = 1$

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

A: To simplify an expression with negative exponents, you can use the following steps:

  1. Rewrite the expression with positive exponents by moving the negative exponent to the other side of the fraction.
  2. Apply the quotient rule to simplify the expression.
  3. Apply the product rule to simplify the expression.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent represents a power of a number, while a negative exponent represents a reciprocal of a power of a number.

For example, $a^m$ represents a power of a number, while $a^{-m}$ represents a reciprocal of a power of a number.

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

A: To simplify an expression with multiple exponents, you can use the following steps:

  1. Apply the product rule to simplify the expression.
  2. Apply the power rule to simplify the expression.
  3. Apply the quotient rule to simplify the expression.

Q: What is the final answer to the expression $(5x){-3}\left(25x{-8}\right)$?

A: The final answer to the expression $(5x){-3}\left(25x{-8}\right)$ is $\frac{1}{5x^{11}}$.

Q: Where can I find more information on simplifying exponents?

A: You can find more information on simplifying exponents on the following websites:

  • Khan Academy: Exponents and Exponential Functions
  • Mathway: Exponents and Exponential Functions
  • Wolfram Alpha: Exponents and Exponential Functions

Conclusion

In this article, we have addressed some of the most frequently asked questions about simplifying exponents. We have provided step-by-step instructions on how to simplify expressions with negative exponents, multiple exponents, and positive exponents. We have also provided the final answer to the expression $(5x){-3}\left(25x{-8}\right)$ and recommended resources for further learning.

Answer Key

The correct answers to the questions are as follows:

  • Q: What is the rule for simplifying exponents? A: The rule for simplifying exponents is the product rule, power rule, quotient rule, and zero exponent rule.
  • Q: How do I simplify an expression with negative exponents? A: To simplify an expression with negative exponents, you can use the following steps: rewrite the expression with positive exponents, apply the quotient rule, and apply the product rule.
  • Q: What is the difference between a positive exponent and a negative exponent? A: A positive exponent represents a power of a number, while a negative exponent represents a reciprocal of a power of a number.
  • Q: How do I simplify an expression with multiple exponents? A: To simplify an expression with multiple exponents, you can use the following steps: apply the product rule, apply the power rule, and apply the quotient rule.
  • Q: What is the final answer to the expression $(5x){-3}\left(25x-8}\right)$? A The final answer to the expression $(5x)^{-3\left(25x^{-8}\right)$ is $\frac{1}{5x^{11}}$.
  • Q: Where can I find more information on simplifying exponents? A: You can find more information on simplifying exponents on the following websites: Khan Academy, Mathway, and Wolfram Alpha.