Simplify. Express Your Answer Using Positive Exponents.$\frac{w^{-1}}{w^6}$

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Introduction

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When dealing with exponents, it's essential to understand the rules for simplifying expressions. In this article, we will focus on simplifying the expression $\frac{w^{-1}}{w^6}$ using positive exponents.

Understanding Exponents

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Exponents are a shorthand way of representing repeated multiplication. For example, $w^3$ can be written as $w \times w \times w$. When we have a negative exponent, it means we are dealing with a reciprocal. In other words, $w^{-1}$ is equal to $\frac{1}{w}$.

Simplifying the Expression

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To simplify the expression $\frac{w^{-1}}{w^6}$, we need to apply the rule for dividing exponents with the same base. When we divide two exponents with the same base, we subtract the exponents. In this case, we have:

$$\frac{w^{-1}}{w^6} = w^{-1-6} = w^{-7}$$

Converting to Positive Exponent

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Now that we have simplified the expression to $w^{-7}$, we need to convert it to a positive exponent. To do this, we can use the rule that states $a^{-n} = \frac{1}{a^n}$. Applying this rule to our expression, we get:

$$w^{-7} = \frac{1}{w^7}$$

Conclusion

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In this article, we simplified the expression $\frac{w^{-1}}{w^6}$ using positive exponents. We applied the rule for dividing exponents with the same base and then converted the negative exponent to a positive exponent using the rule $a^{-n} = \frac{1}{a^n}$. This process is essential in simplifying expressions and solving mathematical problems.

Examples and Practice

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Example 1

Simplify the expression $\frac{w^{-2}}{w^4}$ using positive exponents.

Solution

To simplify the expression, we apply the rule for dividing exponents with the same base:

$$\frac{w^{-2}}{w^4} = w^{-2-4} = w^{-6}$$

Then, we convert the negative exponent to a positive exponent:

$$w^{-6} = \frac{1}{w^6}$$

Example 2

Simplify the expression $\frac{w^{-3}}{w^2}$ using positive exponents.

Solution

To simplify the expression, we apply the rule for dividing exponents with the same base:

$$\frac{w^{-3}}{w^2} = w^{-3-2} = w^{-5}$$

Then, we convert the negative exponent to a positive exponent:

$$w^{-5} = \frac{1}{w^5}$$

Tips and Tricks

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• When dealing with negative exponents, remember that $a^{-n} = \frac{1}{a^n}$.
• When dividing exponents with the same base, subtract the exponents.
• Practice simplifying expressions with negative exponents to become more comfortable with the rules.

Final Thoughts

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Simplifying expressions using positive exponents is an essential skill in mathematics. By applying the rules for dividing exponents with the same base and converting negative exponents to positive exponents, we can simplify complex expressions and solve mathematical problems with ease. Remember to practice regularly and become familiar with the rules to become proficient in simplifying expressions.

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Introduction

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In our previous article, we discussed how to simplify expressions using positive exponents. We covered the rules for dividing exponents with the same base and converting negative exponents to positive exponents. In this article, we will answer some frequently asked questions about simplifying expressions using positive exponents.

Q&A

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

A: When dividing exponents with the same base, we subtract the exponents. For example, $\frac{w^3}{w^2} = w^{3-2} = w^1$.

Q: How do I convert a negative exponent to a positive exponent?

A: To convert a negative exponent to a positive exponent, we use the rule $a^{-n} = \frac{1}{a^n}$. For example, $w^{-3} = \frac{1}{w^3}$.

Q: Can I simplify an expression with a negative exponent by multiplying it by a fraction?

A: Yes, you can simplify an expression with a negative exponent by multiplying it by a fraction. For example, $w^{-3} = \frac{1}{w^3} = \frac{w^0}{w^3} = \frac{1}{w^3}$.

Q: What is the difference between $w^{-3}$ and $\frac{1}{w^3}$?

A: $w^{-3}$ and $\frac{1}{w^3}$ are equivalent expressions. $w^{-3}$ is a negative exponent, while $\frac{1}{w^3}$ is a fraction with a positive exponent.

Q: Can I simplify an expression with a negative exponent by using the rule $a^{-n} = \frac{1}{a^n}$?

A: Yes, you can simplify an expression with a negative exponent by using the rule $a^{-n} = \frac{1}{a^n}$. For example, $w^{-3} = \frac{1}{w^3}$.

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

A: When multiplying exponents with the same base, we add the exponents. For example, $w^3 \times w^2 = w^{3+2} = w^5$.

Q: Can I simplify an expression with a negative exponent by using the rule $a^{-n} = \frac{1}{a^n}$ and then multiplying it by a fraction?

A: Yes, you can simplify an expression with a negative exponent by using the rule $a^{-n} = \frac{1}{a^n}$ and then multiplying it by a fraction. For example, $w^{-3} = \frac{1}{w^3} = \frac{w^0}{w^3} = \frac{1}{w^3}$.

Examples and Practice

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Example 1

Simplify the expression $\frac{w^{-2}}{w^4}$ using positive exponents.

Solution

To simplify the expression, we apply the rule for dividing exponents with the same base:

$$\frac{w^{-2}}{w^4} = w^{-2-4} = w^{-6}$$

Then, we convert the negative exponent to a positive exponent:

$$w^{-6} = \frac{1}{w^6}$$

Example 2

Simplify the expression $\frac{w^{-3}}{w^2}$ using positive exponents.

Solution

To simplify the expression, we apply the rule for dividing exponents with the same base:

$$\frac{w^{-3}}{w^2} = w^{-3-2} = w^{-5}$$

Then, we convert the negative exponent to a positive exponent:

$$w^{-5} = \frac{1}{w^5}$$

Tips and Tricks

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• When dealing with negative exponents, remember that $a^{-n} = \frac{1}{a^n}$.
• When dividing exponents with the same base, subtract the exponents.
• Practice simplifying expressions with negative exponents to become more comfortable with the rules.
• Use the rule $a^{-n} = \frac{1}{a^n}$ to convert negative exponents to positive exponents.
• Use the rule $a^{-n} = \frac{1}{a^n}$ and then multiply by a fraction to simplify expressions with negative exponents.

Final Thoughts

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Simplifying expressions using positive exponents is an essential skill in mathematics. By applying the rules for dividing exponents with the same base and converting negative exponents to positive exponents, we can simplify complex expressions and solve mathematical problems with ease. Remember to practice regularly and become familiar with the rules to become proficient in simplifying expressions.