Simplify The Expression. Write The Answer With Positive Exponents Only. Assume The Variable Represents A Nonzero Real Number.$\frac{r^{-7}}{r^6} =$ $\square$

Feb 28, 2025 by ADMIN 158 views

Simplify the Expression: A Step-by-Step Guide to Positive Exponents

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Introduction

In algebra, simplifying expressions with exponents is a crucial skill that helps us solve equations and manipulate mathematical statements. When dealing with negative exponents, it's essential to rewrite them as positive exponents to make the expression more manageable. In this article, we'll explore how to simplify the expression $\frac{r^{-7}}{r^6}$ and write the answer with positive exponents only.

Understanding Negative Exponents

Before we dive into simplifying the expression, let's quickly review what negative exponents mean. A negative exponent indicates that the base is being raised to a power that is the reciprocal of the given exponent. In other words, $a^{-n} = \frac{1}{a^n}$ . This concept is crucial in simplifying expressions with negative exponents.

Simplifying the Expression

Now, let's focus on simplifying the expression $\frac{r^{-7}}{r^6}$ . To do this, we'll use the rule for dividing exponents with the same base, which states that $\frac{a^m}{a^n} = a^{m-n}$ . Applying this rule to our expression, we get:

\frac{r^{-7}}{r^6} = r^{-7-6} = r^{-13}

Rewriting Negative Exponents as Positive Exponents

As mentioned earlier, we want to rewrite the expression with positive exponents only. To do this, we'll use the rule for negative exponents, which states that $a^{-n} = \frac{1}{a^n}$ . Applying this rule to our expression, we get:

r^{-13} = \frac{1}{r^{13}}

Simplifying the Expression Further

Now that we have the expression with positive exponents, we can simplify it further by applying the rule for dividing exponents with the same base. However, in this case, we don't need to simplify the expression further because we've already achieved our goal of rewriting it with positive exponents.

Conclusion

In conclusion, simplifying the expression $\frac{r^{-7}}{r^6}$ and rewriting it with positive exponents only requires a clear understanding of negative exponents and the rules for dividing exponents with the same base. By applying these rules, we can rewrite the expression as $\frac{1}{r^{13}}$ , which is the final answer.

Frequently Asked Questions

Q: What is the rule for dividing exponents with the same base?

A: The rule for dividing exponents with the same base states that $\frac{a^m}{a^n} = a^{m-n}$ .

Q: How do I rewrite a negative exponent as a positive exponent?

A: To rewrite a negative exponent as a positive exponent, use the rule $a^{-n} = \frac{1}{a^n}$ .

Q: Can I simplify the expression further?

A: In this case, we've already achieved our goal of rewriting the expression with positive exponents, so we don't need to simplify it further.

Final Answer

The final answer is $\boxed{\frac{1}{r^{13}}}$ .

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Introduction

In our previous article, we explored how to simplify the expression $\frac{r^{-7}}{r^6}$ and write the answer with positive exponents only. In this article, we'll continue to provide more information and answer frequently asked questions about simplifying expressions with exponents.

Q&A: Simplifying Expressions with Exponents

Q: What is the rule for simplifying expressions with exponents?

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

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

A: To simplify an expression with a negative exponent, use the rule $a^{-n} = \frac{1}{a^n}$ . This will help you rewrite the expression with a positive exponent.

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

A: Yes, you can simplify an expression with a zero exponent. When the exponent is zero, the expression is equal to 1. In other words, $a^0 = 1$ .

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

A: To simplify an expression with a fractional exponent, use the rule $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ . This will help you rewrite the expression with a positive exponent.

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

A: Yes, you can simplify an expression with a negative base. When the base is negative, the exponent will also be negative. In other words, $(-a)^n = a^n$ if $n$ is even, and $(-a)^n = -a^n$ if $n$ is odd.

Q: How do I simplify an expression with a variable base?

A: To simplify an expression with a variable base, use the rules for exponents that we discussed earlier. Make sure to apply the rules correctly to simplify the expression.

Examples of Simplifying Expressions with Exponents

Example 1: Simplifying an Expression with a Negative Exponent

Simplify the expression $\frac{r^{-3}}{r^2}$ .

Solution:

$\frac{r^{-3}}{r^2} = r^{-3-2} = r^{-5} = \frac{1}{r^5}$

Example 2: Simplifying an Expression with a Zero Exponent

Simplify the expression $r^0$ .

Solution:

$r^0 = 1$

Example 3: Simplifying an Expression with a Fractional Exponent

Simplify the expression $\sqrt[3]{r^2}$ .

Solution:

$\sqrt[3]{r^2} = r^{\frac{2}{3}}$

Conclusion

In conclusion, simplifying expressions with exponents requires a clear understanding of the rules for exponents. By applying these rules, we can simplify expressions with negative exponents, zero exponents, fractional exponents, and variable bases. We hope that this article has provided you with a better understanding of how to simplify expressions with exponents.

Frequently Asked Questions

Q: What is the rule for simplifying expressions with exponents?

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

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

A: To simplify an expression with a negative exponent, use the rule $a^{-n} = \frac{1}{a^n}$ . This will help you rewrite the expression with a positive exponent.

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

A: Yes, you can simplify an expression with a zero exponent. When the exponent is zero, the expression is equal to 1. In other words, $a^0 = 1$ .

Final Answer

The final answer is $\boxed{\frac{1}{r^5}}$ for the first example, $\boxed{1}$ for the second example, and $\boxed{r^{\frac{2}{3}}}$ for the third example.