Which Is The Simplified Form Of $r^{-7} + S^{-12}$?A. $\frac{1}{r^7 S^{12}}$B. $ R R − S 12 R^r - S^{12} R R − S 12 [/tex]C. $\frac{r 7}{s {12}}$D. $\frac{1}{r^7} + \frac{1}{s^{12}}$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us to solve problems more efficiently. When dealing with exponential expressions, it's essential to understand the rules of exponents to simplify them correctly. In this article, we will focus on simplifying the expression $r^{-7} + s^{-12}$ and explore the different options provided.

Understanding Exponents

Before we dive into simplifying the expression, let's review the basics of exponents. An exponent is a small number that is written above and to the right of a number or a variable. It represents the power to which the base is raised. For example, in the expression $r^7$, the exponent 7 indicates that the base r is raised to the power of 7.

Simplifying Negative Exponents

Now, let's focus on simplifying negative exponents. A negative exponent is a fraction with a positive exponent in the denominator. To simplify a negative exponent, we can rewrite it as a fraction with a positive exponent in the numerator and a 1 in the denominator. For example, $r^{-7}$ can be rewritten as $\frac{1}{r^7}$.

Simplifying the Expression

Now that we have a better understanding of negative exponents, let's simplify the expression $r^{-7} + s^{-12}$. Using the rule we learned earlier, we can rewrite each term as a fraction with a positive exponent in the numerator and a 1 in the denominator.

r−7+s−12=1r7+1s12r^{-7} + s^{-12} = \frac{1}{r^7} + \frac{1}{s^{12}}

Evaluating the Options

Now that we have simplified the expression, let's evaluate the options provided.

  • Option A: $\frac{1}{r^7 s^{12}}$
  • Option B: $r^r - s^{12}$
  • Option C: $\frac{r7}{s{12}}$
  • Option D: $\frac{1}{r^7} + \frac{1}{s^{12}}$

Conclusion

Based on our simplification of the expression, we can see that the correct answer is Option D: $\frac{1}{r^7} + \frac{1}{s^{12}}$. This is because we simplified the expression to $\frac{1}{r^7} + \frac{1}{s^{12}}$, which matches Option D.

Final Thoughts

Simplifying exponential expressions is an essential skill in mathematics. By understanding the rules of exponents and applying them correctly, we can simplify complex expressions and solve problems more efficiently. In this article, we focused on simplifying the expression $r^{-7} + s^{-12}$ and explored the different options provided. We hope that this article has provided you with a better understanding of simplifying exponential expressions and has helped you to develop your problem-solving skills.

Common Mistakes to Avoid

When simplifying exponential expressions, there are several common mistakes to avoid.

  • Not understanding the rules of exponents: Make sure you understand the rules of exponents, including the product rule, quotient rule, and power rule.
  • Not applying the rules correctly: Make sure you apply the rules of exponents correctly to simplify the expression.
  • Not checking your work: Make sure you check your work to ensure that the expression is simplified correctly.

Real-World Applications

Simplifying exponential expressions has several real-world applications.

  • Science and engineering: Exponential expressions are used to model population growth, chemical reactions, and electrical circuits.
  • Finance: Exponential expressions are used to calculate compound interest and investment returns.
  • Computer science: Exponential expressions are used to model algorithms and data structures.

Conclusion

Q: What is the rule for simplifying negative exponents?

A: The rule for simplifying negative exponents is to rewrite it as a fraction with a positive exponent in the numerator and a 1 in the denominator. For example, $r^{-7}$ can be rewritten as $\frac{1}{r^7}$.

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

A: To simplify an expression with multiple negative exponents, you can apply the rule for simplifying negative exponents to each term separately. For example, $r^{-7} + s^{-12}$ can be simplified as $\frac{1}{r^7} + \frac{1}{s^{12}}$.

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

A: A positive exponent represents a power to which the base is raised, while a negative exponent represents a fraction with a positive exponent in the denominator. For example, $r^7$ represents a power to which the base r is raised, while $r^{-7}$ represents a fraction with a positive exponent in the denominator.

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

A: To simplify an expression with a negative exponent and a positive exponent, you can apply the rule for simplifying negative exponents to the negative exponent term. For example, $r^7 - r^{-7}$ can be simplified as $r^7 - \frac{1}{r^7}$.

Q: What is the rule for simplifying expressions with the same base?

A: The rule for simplifying expressions with the same base is to add or subtract the exponents. For example, $r^7 + r^7$ can be simplified as $2r^7$.

Q: How do I simplify an expression with a variable in the exponent?

A: To simplify an expression with a variable in the exponent, you can apply the rule for simplifying negative exponents to the variable term. For example, $x^{-7}$ can be simplified as $\frac{1}{x^7}$.

Q: What is the difference between an exponential expression and a polynomial expression?

A: An exponential expression is an expression that contains a base raised to a power, while a polynomial expression is an expression that contains variables and coefficients combined using addition, subtraction, and multiplication. For example, $r^7$ is an exponential expression, while $r^7 + 3r^2$ is a polynomial expression.

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

A: To simplify an expression with multiple terms, you can apply the rules for simplifying negative exponents, adding or subtracting exponents, and combining like terms. For example, $r^7 + r^{-7} + 3r^2$ can be simplified as $r^7 + \frac{1}{r^7} + 3r^2$.

Conclusion

In conclusion, simplifying exponential expressions is an essential skill in mathematics. By understanding the rules of exponents and applying them correctly, we can simplify complex expressions and solve problems more efficiently. We hope that this article has provided you with a better understanding of simplifying exponential expressions and has helped you to develop your problem-solving skills.

Common Mistakes to Avoid

When simplifying exponential expressions, there are several common mistakes to avoid.

  • Not understanding the rules of exponents: Make sure you understand the rules of exponents, including the product rule, quotient rule, and power rule.
  • Not applying the rules correctly: Make sure you apply the rules of exponents correctly to simplify the expression.
  • Not checking your work: Make sure you check your work to ensure that the expression is simplified correctly.

Real-World Applications

Simplifying exponential expressions has several real-world applications.

  • Science and engineering: Exponential expressions are used to model population growth, chemical reactions, and electrical circuits.
  • Finance: Exponential expressions are used to calculate compound interest and investment returns.
  • Computer science: Exponential expressions are used to model algorithms and data structures.

Conclusion

In conclusion, simplifying exponential expressions is an essential skill in mathematics. By understanding the rules of exponents and applying them correctly, we can simplify complex expressions and solve problems more efficiently. We hope that this article has provided you with a better understanding of simplifying exponential expressions and has helped you to develop your problem-solving skills.