18. Simplify:$-\left(3 A B^2\right)^{-3}$Options:A. $\frac{1}{9 A^3 B^6}$B. $-\frac{1}{27 A^3 B^6}$

Understanding the Problem

In this problem, we are given the expression $-\left(3 a b^2\right)^{-3}$ and we need to simplify it. To simplify this expression, we will use the properties of exponents and the rules of negative exponents.

The Rules of Negative Exponents

A negative exponent is a shorthand way of writing a fraction. For example, $a^{-n} = \frac{1}{a^n}$. This means that when we see a negative exponent, we can rewrite it as a fraction with a positive exponent in the denominator.

Simplifying the Expression

Let's start by simplifying the expression inside the parentheses. We have $\left(3 a b^2\right)^{-3}$. Using the rule of negative exponents, we can rewrite this as $\frac{1}{\left(3 a b^2\right)^3}$.

Applying the Power of a Product Rule

The power of a product rule states that $\left(ab\right)^n = a^n b^n$. We can use this rule to simplify the expression inside the parentheses. We have $\left(3 a b^2\right)^3 = 3^3 a^3 b^6$.

Simplifying the Expression Further

Now that we have simplified the expression inside the parentheses, we can simplify the entire expression. We have $\frac{1}{\left(3 a b^2\right)^3} = \frac{1}{3^3 a^3 b^6}$.

Applying the Rule of Negative Exponents Again

We can use the rule of negative exponents again to rewrite the expression as $-\frac{1}{3^3 a^3 b^6}$.

Simplifying the Numerator

The numerator of the expression is $-1$. We can simplify this by multiplying the numerator and denominator by $-1$ to get $\frac{1}{3^3 a^3 b^6}$.

Simplifying the Denominator

The denominator of the expression is $3^3 a^3 b^6$. We can simplify this by evaluating the exponent. We have $3^3 = 27$.

The Final Answer

Now that we have simplified the expression, we can write the final answer. We have $-\frac{1}{27 a^3 b^6}$.

Conclusion

In this problem, we simplified the expression $-\left(3 a b^2\right)^{-3}$ using the properties of exponents and the rules of negative exponents. We started by simplifying the expression inside the parentheses, then applied the power of a product rule, and finally applied the rule of negative exponents again to get the final answer.

Answer Options

There are two answer options for this problem:

• A. $\frac{1}{9 a^3 b^6}$
• B. $-\frac{1}{27 a^3 b^6}$

Which Answer is Correct?

The correct answer is B. $-\frac{1}{27 a^3 b^6}$. This is the final answer we obtained after simplifying the expression.

Why is this Answer Correct?

This answer is correct because we simplified the expression using the properties of exponents and the rules of negative exponents. We started by simplifying the expression inside the parentheses, then applied the power of a product rule, and finally applied the rule of negative exponents again to get the final answer.

What is the Importance of Simplifying Expressions?

Simplifying expressions is an important skill in mathematics because it helps us to:

• Evaluate expressions more easily
• Solve equations more easily
• Understand the properties of exponents and other mathematical concepts
• Make calculations more efficient

Conclusion

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Q: What is the rule for simplifying exponential expressions?

A: The rule for simplifying exponential expressions is to use the properties of exponents and the rules of negative exponents. Specifically, we can use the power of a product rule, the power of a power rule, and the rule of negative exponents to simplify exponential expressions.

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

A: To simplify an expression with a negative exponent, we can use the rule of negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. This means that we can rewrite a negative exponent as a fraction with a positive exponent in the denominator.

Q: What is the power of a product rule?

A: The power of a product rule states that $\left(ab\right)^n = a^n b^n$. This means that when we raise a product to a power, we can raise each factor to that power separately.

Q: How do I apply the power of a product rule?

A: To apply the power of a product rule, we simply raise each factor to the power that is given. For example, if we have $\left(3a\right)^2$, we can apply the power of a product rule to get $3^2 a^2$.

Q: What is the power of a power rule?

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

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

A: To apply the power of a power rule, we simply multiply the exponents. For example, if we have $\left(a^2\right)^3$, we can apply the power of a power rule to get $a^{2\cdot3} = a^6$.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, we can use the power of a product rule and the power of a power rule to simplify each factor separately, and then combine the results.

Q: What is the rule for simplifying expressions with variables in the exponent?

A: The rule for simplifying expressions with variables in the exponent is to use the properties of exponents and the rules of negative exponents. Specifically, we can use the power of a product rule, the power of a power rule, and the rule of negative exponents to simplify expressions with variables in the exponent.

Q: How do I simplify an expression with a variable in the exponent?

A: To simplify an expression with a variable in the exponent, we can use the properties of exponents and the rules of negative exponents. Specifically, we can use the power of a product rule, the power of a power rule, and the rule of negative exponents to simplify the expression.

Q: What is the importance of simplifying expressions?

A: Simplifying expressions is an important skill in mathematics because it helps us to:

• Evaluate expressions more easily
• Solve equations more easily
• Understand the properties of exponents and other mathematical concepts
• Make calculations more efficient

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working through examples and exercises in your textbook or online resources. You can also try simplifying expressions on your own, using the properties of exponents and the rules of negative exponents.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Forgetting to apply the power of a product rule or the power of a power rule
• Forgetting to use the rule of negative exponents
• Not simplifying expressions fully
• Making errors when multiplying or dividing exponents

Q: How can I check my work when simplifying expressions?

A: You can check your work when simplifying expressions by:

• Verifying that you have applied the correct rules
• Checking that your answer is in the simplest form possible
• Using a calculator or online tool to check your answer
• Working through the problem again to make sure you get the same answer.