Which Expression Is Equivalent To $\frac{r^9}{r^3}$?A. $r^3$ B. $r^6$ C. $r^{12}$ D. $r^{27}$

Introduction

When working with exponents, it's essential to understand the rules for simplifying expressions. In this article, we'll explore the concept of equivalent expressions and how to simplify exponents using the quotient rule. We'll apply this knowledge to a specific problem, where we need to find the equivalent expression for $\frac{r^9}{r^3}$.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $r^3$ means $r \times r \times r$. When we have a fraction with exponents, such as $\frac{r^9}{r^3}$, we can simplify it using the quotient rule.

The Quotient Rule

The quotient rule states that when we divide two powers with the same base, we subtract the exponents. In other words, $\frac{a^m}{a^n} = a^{m-n}$. This rule applies to all bases, including variables like $r$.

Applying the Quotient Rule

Now, let's apply the quotient rule to the expression $\frac{r^9}{r^3}$. We can rewrite this expression as $r^{9-3}$, using the quotient rule. Simplifying the exponent, we get $r^6$.

Evaluating the Options

Now that we've simplified the expression, let's evaluate the options:

• A. $r^3$ - This is not the correct answer, as we simplified the expression to $r^6$, not $r^3$.
• B. $r^6$ - This is the correct answer, as we simplified the expression to $r^6$.
• C. $r^{12}$ - This is not the correct answer, as we simplified the expression to $r^6$, not $r^{12}$.
• D. $r^{27}$ - This is not the correct answer, as we simplified the expression to $r^6$, not $r^{27}$.

Conclusion

In this article, we explored the concept of equivalent expressions and how to simplify exponents using the quotient rule. We applied this knowledge to a specific problem, where we needed to find the equivalent expression for $\frac{r^9}{r^3}$. By following the quotient rule, we simplified the expression to $r^6$. This demonstrates the importance of understanding exponents and how to simplify expressions using the quotient rule.

Additional Examples

To reinforce your understanding of the quotient rule, let's consider a few additional examples:

• $\frac{r^4}{r^2} = r^{4-2} = r^2$
• $\frac{r^6}{r^3} = r^{6-3} = r^3$
• $\frac{r^8}{r^4} = r^{8-4} = r^4$

These examples demonstrate how the quotient rule can be applied to simplify expressions with exponents.

Common Mistakes

When working with exponents, it's easy to make mistakes. Here are a few common errors to watch out for:

• Forgetting to apply the quotient rule when dividing powers with the same base.
• Not simplifying the exponent correctly.
• Not evaluating the options correctly.

By being aware of these common mistakes, you can avoid them and ensure that your calculations are accurate.

Final Thoughts

Q&A: Simplifying Exponents

Q: What is the quotient rule in exponents?

A: The quotient rule states that when we divide two powers with the same base, we subtract the exponents. In other words, $\frac{a^m}{a^n} = a^{m-n}$.

Q: How do I apply the quotient rule to simplify an expression?

A: To apply the quotient rule, simply subtract the exponents of the two powers with the same base. For example, $\frac{r^9}{r^3} = r^{9-3} = r^6$.

Q: What if I have a fraction with exponents and the bases are different?

A: If the bases are different, we cannot apply the quotient rule. In this case, we need to simplify the expression using other rules, such as the product rule or the power rule.

Q: Can I simplify an expression with a negative exponent?

A: Yes, we can simplify an expression with a negative exponent by applying the quotient rule. For example, $\frac{r^3}{r^{-2}} = r^{3-(-2)} = r^5$.

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

A: An expression with a zero exponent is equal to 1. For example, $r^0 = 1$.

Q: Can I simplify an expression with a variable base and a variable exponent?

A: Yes, we can simplify an expression with a variable base and a variable exponent by applying the quotient rule. For example, $\frac{x^5}{x^2} = x^{5-2} = x^3$.

Q: What if I have a fraction with exponents and the bases are the same, but the exponents are different?

A: If the bases are the same, but the exponents are different, we can apply the quotient rule to simplify the expression. For example, $\frac{r^4}{r^2} = r^{4-2} = r^2$.

Q: Can I simplify an expression with a fraction and a negative exponent?

A: Yes, we can simplify an expression with a fraction and a negative exponent by applying the quotient rule. For example, $\frac{r^{-3}}{r^2} = r^{-3-2} = r^{-5}$.

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

A: We can simplify an expression with a fraction and a variable exponent by applying the quotient rule. For example, $\frac{x^3}{x^2} = x^{3-2} = x^1 = x$.

Q: Can I simplify an expression with a fraction and a zero exponent?

A: Yes, we can simplify an expression with a fraction and a zero exponent by applying the quotient rule. For example, $\frac{r^3}{r^0} = r^{3-0} = r^3$.

Conclusion

In this article, we've covered some common questions and answers related to simplifying exponents using the quotient rule. By understanding the quotient rule and how to apply it, you can simplify expressions with exponents and solve problems with ease. Remember to always evaluate the options carefully and check your work to ensure that your calculations are accurate. With practice and patience, you'll become proficient in simplifying exponents and solving problems with confidence.