Nadine And Calvin Are Simplifying The Expression ( R − 5 S − 3 R 8 S − 2 ) − 4 \left(\frac{r^{-5} S^{-3}}{r^8 S^{-2}}\right)^{-4} ( R 8 S − 2 R − 5 S − 3 ​ ) − 4 .Nadine Claims The First Step To Simplify The Expression Is To Raise The Numerator And Denominator To The Power Of 4 To Get $\frac{r^{-20}

Introduction

In mathematics, simplifying exponential expressions is a crucial skill that helps us solve complex problems and understand the underlying concepts. Nadine and Calvin are working on simplifying the expression $\left(\frac{r^{-5} s^{-3}}{r^8 s^{-2}}\right)^{-4}$. In this article, we will guide you through the step-by-step process of simplifying this expression and provide you with a deeper understanding of the underlying concepts.

Understanding Exponents

Before we dive into the simplification process, let's review the basics of exponents. An exponent is a small number that is written above and to the right of a base number. It tells us how many times to multiply the base number by itself. For example, $r^3$ means $r$ multiplied by itself three times, or $r \times r \times r$. When we have a negative exponent, it means we are taking the reciprocal of the base number. For example, $r^{-3}$ means $\frac{1}{r^3}$.

Simplifying the Expression

Now that we have a good understanding of exponents, let's simplify the expression $\left(\frac{r^{-5} s^{-3}}{r^8 s^{-2}}\right)^{-4}$. Nadine claims that the first step is to raise the numerator and denominator to the power of 4 to get $\frac{r^{-20} s^{-12}}{r^{32} s^{-8}}$. However, this is not the correct approach.

The Correct Approach

To simplify the expression, we need to follow the order of operations (PEMDAS). The first step is to evaluate the exponent outside the parentheses. When we have a negative exponent outside the parentheses, we need to take the reciprocal of the expression inside the parentheses. In this case, we have $\left(\frac{r^{-5} s^{-3}}{r^8 s^{-2}}\right)^{-4}$. To simplify this expression, we need to raise the numerator and denominator to the power of -4, not 4.

Raising to the Power of -4

When we raise the numerator and denominator to the power of -4, we get $\frac{(r^{-5} s^{-3})^{-4}}{(r^8 s^{-2})^{-4}}$. Using the rule of exponents that states $(a^m)^n = a^{mn}$, we can simplify this expression to $\frac{r^{20} s^{12}}{r^{-32} s^{8}}$.

Simplifying Further

Now that we have raised the numerator and denominator to the power of -4, we can simplify the expression further. We can combine like terms in the numerator and denominator. In the numerator, we have $r^{20}$ and $s^{12}$. In the denominator, we have $r^{-32}$ and $s^{8}$. We can combine these terms by adding their exponents. This gives us $\frac{r^{20+32} s^{12-8}}{1}$.

Final Simplification

After combining like terms, we are left with $\frac{r^{52} s^{4}}{1}$. Since we are dividing by 1, we can simplify this expression to $r^{52} s^{4}$.

Conclusion

In this article, we have simplified the expression $\left(\frac{r^{-5} s^{-3}}{r^8 s^{-2}}\right)^{-4}$ using the correct approach. We raised the numerator and denominator to the power of -4, combined like terms, and simplified the expression further. The final simplified expression is $r^{52} s^{4}$. We hope this article has provided you with a deeper understanding of simplifying exponential expressions and has helped you develop your problem-solving skills.

Common Mistakes to Avoid

When simplifying exponential expressions, there are several common mistakes to avoid. One of the most common mistakes is raising the numerator and denominator to the wrong power. In this case, Nadine raised the numerator and denominator to the power of 4, instead of -4. This led to an incorrect simplification of the expression.

Tips and Tricks

When simplifying exponential expressions, it's essential to follow the order of operations (PEMDAS). This means evaluating the exponent outside the parentheses first, then raising the numerator and denominator to the correct power. Additionally, be careful when combining like terms, as this can lead to errors if not done correctly.

Practice Problems

To practice simplifying exponential expressions, try the following problems:

• Simplify the expression $\left(\frac{a^3 b^2}{a^5 b^4}\right)^{-2}$
• Simplify the expression $\left(\frac{x^{-2} y^3}{x^4 y^{-5}}\right)^{-3}$
• Simplify the expression $\left(\frac{z^6 w^4}{z^2 w^3}\right)^{-1}$

Conclusion

Introduction

In our previous article, we explored the process of simplifying exponential expressions. We discussed the importance of following the order of operations (PEMDAS) and raising the numerator and denominator to the correct power. In this article, we will provide a Q&A guide to help you better understand the concepts and apply them to real-world problems.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

• Parentheses: Evaluate expressions inside parentheses first.
• Exponents: Evaluate any exponential expressions next.
• Multiplication and Division: Evaluate any multiplication and division operations from left to right.
• Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

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

A: When we have a negative exponent, we need to take the reciprocal of the base number. For example, $r^{-3}$ means $\frac{1}{r^3}$. To simplify an exponential expression with a negative exponent, we need to raise the numerator and denominator to the correct power.

Q: What is the rule for raising a power to a power?

A: The rule for raising a power to a power is $(a^m)^n = a^{mn}$. This means that when we raise a power to a power, we multiply the exponents.

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

A: When we have multiple exponents, we need to combine them by adding their exponents. For example, $r^3 s^2$ means $r^3 \times s^2$, or $r^3 s^{2+0}$.

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

A: A positive exponent means that we are multiplying the base number by itself a certain number of times. For example, $r^3$ means $r \times r \times r$. A negative exponent means that we are taking the reciprocal of the base number. For example, $r^{-3}$ means $\frac{1}{r^3}$.

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

A: When we have a zero exponent, the expression is equal to 1. For example, $r^0$ means $1$.

Q: What is the rule for multiplying exponential expressions?

A: The rule for multiplying exponential expressions is $a^m \times a^n = a^{m+n}$. This means that when we multiply exponential expressions with the same base, we add their exponents.

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

A: When we have a fraction exponent, we need to raise the numerator and denominator to the correct power. For example, $\left(\frac{r^3}{s^2}\right)^{-2}$ means $\frac{r^{-6}}{s^4}$.

Q: What is the rule for dividing exponential expressions?

A: The rule for dividing exponential expressions is $\frac{a^m}{a^n} = a^{m-n}$. This means that when we divide exponential expressions with the same base, we subtract their exponents.

Conclusion

In this Q&A guide, we have covered the basics of simplifying exponential expressions. We have discussed the order of operations (PEMDAS), raising a power to a power, and combining multiple exponents. We have also covered the rules for multiplying and dividing exponential expressions. By following these rules and practicing with real-world problems, you will become proficient in simplifying exponential expressions and develop your problem-solving skills.

Practice Problems

To practice simplifying exponential expressions, try the following problems:

• Simplify the expression $\left(\frac{a^3 b^2}{a^5 b^4}\right)^{-2}$
• Simplify the expression $\left(\frac{x^{-2} y^3}{x^4 y^{-5}}\right)^{-3}$
• Simplify the expression $\left(\frac{z^6 w^4}{z^2 w^3}\right)^{-1}$

Additional Resources

For more information on simplifying exponential expressions, check out the following resources:

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

Conclusion

In conclusion, simplifying exponential expressions is a crucial skill that requires attention to detail and a deep understanding of the underlying concepts. By following the order of operations (PEMDAS) and raising the numerator and denominator to the correct power, we can simplify complex expressions and develop our problem-solving skills. We hope this Q&A guide has provided you with a deeper understanding of simplifying exponential expressions and has helped you develop your problem-solving skills.