Use The Power Of A Power Rule To Simplify The Following Expressions Involving Monomials As Much As Possible.$ \left(-9 X^2 Y^6 Z\right)^2 = $ $ \square $

Feb 28, 2025 by ADMIN 155 views

**Simplifying Expressions with the Power Rule**

Introduction

In algebra, the power rule is a fundamental concept used to simplify expressions involving exponents and monomials. It states that when a monomial is raised to a power, we can multiply the exponent by the power to which the monomial is raised. In this article, we will use the power rule to simplify the expression $\left(-9 x^2 y^6 z\right)^2$ as much as possible.

Understanding the Power Rule

The power rule is a simple yet powerful tool for simplifying expressions involving exponents and monomials. It states that when a monomial is raised to a power, we can multiply the exponent by the power to which the monomial is raised. For example, if we have the expression $(2x^3)^4$ , we can use the power rule to simplify it as follows:

$(2x^3)^4 = 2^4 \cdot x^{3 \cdot 4} = 16x^{12}$

As we can see, the power rule allows us to simplify complex expressions by multiplying the exponent by the power to which the monomial is raised.

Applying the Power Rule to the Given Expression

Now that we have a good understanding of the power rule, let's apply it to the given expression $\left(-9 x^2 y^6 z\right)^2$ . To simplify this expression, we need to multiply the exponent of each monomial by the power to which the expression is raised.

$\left(-9 x^2 y^6 z\right)^2 = (-9)^2 \cdot (x^2)^2 \cdot (y^6)^2 \cdot z^2$

Using the power rule, we can simplify each term as follows:

$(-9)^2 = 81$ $(x^2)^2 = x^4$ $(y^6)^2 = y^{12}$ $z^2 = z^2$

Now that we have simplified each term, we can combine them to get the final simplified expression.

Simplifying the Expression

Now that we have simplified each term, we can combine them to get the final simplified expression.

$\left(-9 x^2 y^6 z\right)^2 = 81x^4y^{12}z^2$

As we can see, the power rule has allowed us to simplify the given expression by multiplying the exponent of each monomial by the power to which the expression is raised.

Conclusion

In this article, we have used the power rule to simplify the expression $\left(-9 x^2 y^6 z\right)^2$ as much as possible. We have seen how the power rule can be used to simplify complex expressions by multiplying the exponent by the power to which the monomial is raised. By applying the power rule, we have been able to simplify the given expression and arrive at the final simplified form.

Common Mistakes to Avoid

When applying the power rule, there are a few common mistakes to avoid. These include:

Forgetting to multiply the exponent by the power: This is the most common mistake when applying the power rule. Make sure to multiply the exponent by the power to which the monomial is raised.
Not simplifying each term separately: When applying the power rule, make sure to simplify each term separately before combining them.
Not checking the final expression for errors: Before finalizing the simplified expression, make sure to check it for errors.

Practice Problems

To practice applying the power rule, try simplifying the following expressions:

$(3x^2y^3)^4$
$(2x^3y^2)^5$
$(x^2y^4z^3)^6$

Answer Key

$(3x^2y^3)^4 = 3^4 \cdot x^{2 \cdot 4} \cdot y^{3 \cdot 4} = 81x^8y^{12}$
$(2x^3y^2)^5 = 2^5 \cdot x^{3 \cdot 5} \cdot y^{2 \cdot 5} = 32x^{15}y^{10}$
$(x^2y^4z^3)^6 = x^{2 \cdot 6} \cdot y^{4 \cdot 6} \cdot z^{3 \cdot 6} = x^{12}y^{24}z^{18}$
Power Rule Q&A ==================

Q: What is the power rule in algebra?

A: The power rule is a fundamental concept in algebra that states that when a monomial is raised to a power, we can multiply the exponent by the power to which the monomial is raised.

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

A: To apply the power rule, simply multiply the exponent of each monomial by the power to which the expression is raised. For example, if we have the expression $(2x^3)^4$ , we can use the power rule to simplify it as follows:

$(2x^3)^4 = 2^4 \cdot x^{3 \cdot 4} = 16x^{12}$

Q: What are some common mistakes to avoid when applying the power rule?

A: Some common mistakes to avoid when applying the power rule include:

Forgetting to multiply the exponent by the power
Not simplifying each term separately
Not checking the final expression for errors

Q: Can I use the power rule to simplify expressions with negative exponents?

A: Yes, you can use the power rule to simplify expressions with negative exponents. For example, if we have the expression $(x^{-2})^3$ , we can use the power rule to simplify it as follows:

$(x^{-2})^3 = x^{-2 \cdot 3} = x^{-6}$

Q: Can I use the power rule to simplify expressions with fractional exponents?

A: Yes, you can use the power rule to simplify expressions with fractional exponents. For example, if we have the expression $(x^{1/2})^3$ , we can use the power rule to simplify it as follows:

$(x^{1/2})^3 = x^{(1/2) \cdot 3} = x^{3/2}$

Q: How do I simplify expressions with multiple terms?

A: To simplify expressions with multiple terms, simply apply the power rule to each term separately and then combine the results. For example, if we have the expression $(2x^3y^2)^4$ , we can use the power rule to simplify it as follows:

$(2x^3y^2)^4 = 2^4 \cdot x^{3 \cdot 4} \cdot y^{2 \cdot 4} = 16x^{12}y^8$

Q: Can I use the power rule to simplify expressions with variables in the exponent?

A: Yes, you can use the power rule to simplify expressions with variables in the exponent. For example, if we have the expression $(x^2y^3)^4$ , we can use the power rule to simplify it as follows:

$(x^2y^3)^4 = x^{2 \cdot 4} \cdot y^{3 \cdot 4} = x^8y^{12}$

Q: How do I check my work when simplifying expressions with the power rule?

A: To check your work when simplifying expressions with the power rule, simply plug in a value for the variable and evaluate the expression. For example, if we have the expression $(2x^3)^4$ , we can plug in $x = 2$ and evaluate the expression as follows:

$(2(2)^3)^4 = (2 \cdot 8)^4 = 16^4 = 65536$

We can then check our work by plugging in $x = 2$ into the original expression and evaluating it as follows:

$(2x^3)^4 = (2 \cdot 2^3)^4 = (2 \cdot 8)^4 = 16^4 = 65536$

Since the two expressions are equal, we can be confident that our work is correct.

Conclusion

In this article, we have answered some common questions about the power rule in algebra. We have seen how to apply the power rule to simplify expressions, how to avoid common mistakes, and how to check our work. By following these tips and practicing with examples, you can become more confident and proficient in using the power rule to simplify expressions.