Simplifying the Negative Exponent

Oct 13, 2025 by windailys.com 137 views

Understanding the Problem

When dealing with exponents, it's essential to remember the rules of exponentiation. In this case, we're given the expression ${$ (-5x)^5$}$ and asked to simplify it using only positive exponents. To do this, we'll need to apply the power rule of exponents, which states that ${(a^m)^n = a^{m \cdot n}}$ .

Applying the Power Rule

Using the power rule, we can rewrite the given expression as ${(-5x)^5 = (-5)^5 \cdot x^5}$ . Now, we need to simplify the expression ${(-5)^5}$ . To do this, we'll need to apply the rule for negative exponents, which states that ${(-a)^n = a^n}$ if ${n}$ is even, and ${(-a)^n = -a^n}$ if ${n}$ is odd.

Simplifying the Negative Exponent

Since ${5}$ is an odd number, we can rewrite the expression ${(-5)^5}$ as ${-5^5}$ . Now, we can simplify the expression ${-5^5}$ by applying the rule for negative exponents. We get ${-5^5 = -3125}$ .

Simplifying the Expression

Now that we've simplified the expression ${(-5)^5}$ , we can rewrite the original expression as ${-3125x^5}$ . This is the simplified expression using only positive exponents.

Conclusion

In conclusion, we've simplified the expression ${$ (-5x)^5$}$ using only positive exponents. The simplified expression is ${-3125x^5}$ . We applied the power rule of exponents and the rule for negative exponents to simplify the expression.

Key Takeaways

The power rule of exponents states that ${(a^m)^n = a^{m \cdot n}}$ .
The rule for negative exponents states that ${(-a)^n = a^n}$ if ${n}$ is even, and ${(-a)^n = -a^n}$ if ${n}$ is odd.
To simplify an expression with a negative exponent, we can rewrite it as a positive exponent by applying the rule for negative exponents.

Example Problems

Simplify the expression ${(-3x)^4}$ .
Simplify the expression ${(-2y)^6}$ .

Solution to Example Problems

To simplify the expression ${(-3x)^4}$ , we can apply the power rule of exponents and the rule for negative exponents. We get ${(-3x)^4 = (-3)^4 \cdot x^4 = 81x^4}$ .
To simplify the expression ${(-2y)^6}$ , we can apply the power rule of exponents and the rule for negative exponents. We get ${(-2y)^6 = (-2)^6 \cdot y^6 = 64y^6}$ .

Practice Problems

Simplify the expression ${(-4z)^3}$ .
Simplify the expression ${(-5w)^2}$ .

Answer Key

${(-4z)^3 = -64z^3}$
${(-5w)^2 = 25w^2}$

Final Thoughts

$In this article, we've simplified the expression ${$(-5x)^5\$}$ using only positive exponents. We applied the power rule of exponents and the rule for negative exponents to simplify the expression. We also provided example problems and their solutions to help reinforce the concepts learned in this article.<br/> **Simplify ${$(-5x)^5\$}$ Q&A**$

Q: What is the power rule of exponents?

A: The power rule of exponents states that ${(a^m)^n = a^{m \cdot n}}$ . This means that when we raise a power to another power, we multiply the exponents.

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

A: To simplify an expression with a negative exponent, we can rewrite it as a positive exponent by applying the rule for negative exponents. The rule states that ${(-a)^n = a^n}$ if ${n}$ is even, and ${(-a)^n = -a^n}$ if ${n}$ is odd.

Q: What is the rule for negative exponents?

A: The rule for negative exponents states that ${(-a)^n = a^n}$ if ${n}$ is even, and ${(-a)^n = -a^n}$ if ${n}$ is odd.

Q: How do I apply the power rule of exponents?

A: To apply the power rule of exponents, we simply multiply the exponents. For example, ${(a^m)^n = a^{m \cdot n}}$ .

Q: Can you give an example of how to simplify an expression using the power rule of exponents?

A: Let's say we want to simplify the expression ${(-3x)^4}$ . We can apply the power rule of exponents by multiplying the exponents: ${(-3x)^4 = (-3)^4 \cdot x^4 = 81x^4}$ .

Q: What if the exponent is negative? How do I simplify the expression?

A: If the exponent is negative, we can rewrite it as a positive exponent by applying the rule for negative exponents. For example, ${(-3x)^{-4} = \frac{1}{(-3x)^4} = \frac{1}{81x^4}}$ .

Q: Can you give another example of how to simplify an expression using the power rule of exponents?

A: Let's say we want to simplify the expression ${(-2y)^6}$ . We can apply the power rule of exponents by multiplying the exponents: ${(-2y)^6 = (-2)^6 \cdot y^6 = 64y^6}$ .

Q: What if the expression has multiple variables? How do I simplify it?

A: If the expression has multiple variables, we can simplify it by applying the power rule of exponents to each variable separately. For example, ${(-3x)^4y^2 = (-3)^4 \cdot x^4 \cdot y^2 = 81x^4y^2}$ .

Q: Can you give a final example of how to simplify an expression using the power rule of exponents?

A: Let's say we want to simplify the expression ${(-4z)^3}$ . We can apply the power rule of exponents by multiplying the exponents: ${(-4z)^3 = (-4)^3 \cdot z^3 = -64z^3}$ .

Conclusion

In this article, we've answered some common questions about simplifying expressions using the power rule of exponents. We've covered topics such as the power rule of exponents, the rule for negative exponents, and how to apply the power rule of exponents to simplify expressions. We've also provided example problems and their solutions to help reinforce the concepts learned in this article.