Simplify The Expression To Include Only Positive Exponents. Evaluate Powers Where Appropriate.$\left(\frac{2 A^2 B^{-1}}{3 A^{-2} B^4}\right)^3 = \square$

Introduction

In this article, we will simplify the given expression to include only positive exponents. We will also evaluate powers where appropriate. The expression given is $\left(\frac{2 a^2 b^{-1}}{3 a^{-2} b^4}\right)^3$. Our goal is to simplify this expression and present it in a form that includes only positive exponents.

Step 1: Simplify the Expression Inside the Parentheses

To simplify the expression inside the parentheses, we need to apply the rules of exponents. We know that when we divide two powers with the same base, we subtract the exponents. In this case, we have $\frac{2 a^2 b^{-1}}{3 a^{-2} b^4}$. We can simplify this expression by applying the rule of dividing powers with the same base.

\frac{2 a^2 b^{-1}}{3 a^{-2} b^4} = \frac{2}{3} a^{2-(-2)} b^{-1-4} = \frac{2}{3} a^4 b^{-5}

Step 2: Apply the Power Rule

Now that we have simplified the expression inside the parentheses, we can apply the power rule. The power rule states that when we raise a power to another power, we multiply the exponents. In this case, we have $\left(\frac{2}{3} a^4 b^{-5}\right)^3$. We can apply the power rule to simplify this expression.

\left(\frac{2}{3} a^4 b^{-5}\right)^3 = \left(\frac{2}{3}\right)^3 (a^4)^3 (b^{-5})^3 = \frac{8}{27} a^{12} b^{-15}

Step 3: Simplify the Expression to Include Only Positive Exponents

Now that we have applied the power rule, we can simplify the expression to include only positive exponents. We know that when we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. In this case, we have $b^{-15}$. We can rewrite this as $\frac{1}{b^{15}}$.

\frac{8}{27} a^{12} b^{-15} = \frac{8}{27} a^{12} \frac{1}{b^{15}} = \frac{8 a^{12}}{27 b^{15}}

Conclusion

In this article, we simplified the given expression to include only positive exponents. We applied the rules of exponents and the power rule to simplify the expression. We also evaluated powers where appropriate. The final simplified expression is $\frac{8 a^{12}}{27 b^{15}}$.

Final Answer

Introduction

In our previous article, we simplified the given expression to include only positive exponents. We applied the rules of exponents and the power rule to simplify the expression. In this article, we will answer some frequently asked questions related to simplifying expressions with exponents.

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

A: When we divide two powers with the same base, we subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Q: What is the power rule?

A: The power rule states that when we raise a power to another power, we multiply the exponents. For example, $(a^m)^n = a^{mn}$.

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

A: When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, $a^{-m} = \frac{1}{a^m}$.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the base is taken to a power of the reciprocal. For example, $a^m$ indicates that $a$ is raised to the power of $m$, while $a^{-m}$ indicates that $a$ is taken to the power of the reciprocal of $m$.

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

A: To simplify an expression with multiple exponents, we need to apply the rules of exponents. We can use the product rule, which states that when we multiply two powers with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.

Q: What is the difference between a power and an exponent?

A: A power is the result of raising a base to a certain exponent. For example, $a^m$ is a power, while $m$ is the exponent. An exponent is a number that is raised to a power.

Q: How do we evaluate powers where appropriate?

A: To evaluate powers where appropriate, we need to apply the rules of exponents. We can use the power rule, which states that when we raise a power to another power, we multiply the exponents. For example, $(a^m)^n = a^{mn}$.

Q: What is the final simplified expression for the given problem?

A: The final simplified expression for the given problem is $\frac{8 a^{12}}{27 b^{15}}$.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions with exponents. We covered topics such as the rule for dividing powers with the same base, the power rule, and how to simplify expressions with multiple exponents. We also discussed the difference between a positive exponent and a negative exponent, and how to evaluate powers where appropriate.

Final Answer

The final answer is $\boxed{\frac{8 a^{12}}{27 b^{15}}}$.