Simplify The Expression: ${ \frac{\left(3 A^6 B 3\right) 2}{-9 A^{12} B^4} }$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. In this article, we will focus on simplifying a specific expression involving exponents and fractions. The given expression is $\frac{\left(3 a^6 b^3\right)^2}{-9 a^{12} b^4}$. Our goal is to simplify this expression by applying the rules of exponents and fractions.

Understanding Exponents

Before we dive into simplifying the expression, let's review the rules of exponents. When we have a power raised to another power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$. This rule will be essential in simplifying the given expression.

Simplifying the Expression

To simplify the expression, we will start by applying the rule of exponents to the numerator. We have $\left(3 a^6 b^3\right)^2$, which can be rewritten as $3^2 \cdot a^{6 \cdot 2} \cdot b^{3 \cdot 2}$. Using the rule of exponents, we get $9 a^{12} b^6$.

Applying the Rule of Exponents to the Denominator

Now, let's focus on the denominator. We have $-9 a^{12} b^4$. We can rewrite this as $-9 \cdot a^{12} \cdot b^4$.

Simplifying the Fraction

Now that we have simplified the numerator and denominator, we can rewrite the expression as $\frac{9 a^{12} b^6}{-9 a^{12} b^4}$. To simplify this fraction, we can cancel out the common factors in the numerator and denominator. We have $9$ in the numerator and $-9$ in the denominator, which can be canceled out. We also have $a^{12}$ in both the numerator and denominator, which can be canceled out. Finally, we have $b^6$ in the numerator and $b^4$ in the denominator, which can be rewritten as $b^{6-4}$.

Final Simplification

After canceling out the common factors, we are left with $\frac{b^2}{-1}$. This can be rewritten as $-b^2$.

Conclusion

In this article, we simplified the expression $\frac{\left(3 a^6 b^3\right)^2}{-9 a^{12} b^4}$ by applying the rules of exponents and fractions. We started by simplifying the numerator and denominator separately, and then we canceled out the common factors to get the final simplified expression. The final simplified expression is $-b^2$.

Tips and Tricks

• When simplifying expressions involving exponents, make sure to apply the rules of exponents carefully.
• When simplifying fractions, make sure to cancel out the common factors in the numerator and denominator.
• Practice simplifying expressions involving exponents and fractions to become more comfortable with the rules and techniques.

Common Mistakes to Avoid

• When simplifying expressions involving exponents, make sure to apply the rules of exponents carefully. For example, $(a^m)^n = a^{m \cdot n}$, not $a^{m + n}$.
• When simplifying fractions, make sure to cancel out the common factors in the numerator and denominator. For example, $\frac{a^m}{a^n} = a^{m-n}$, not $a^m$.

Real-World Applications

Simplifying expressions involving exponents and fractions has many real-world applications. For example, in physics, we often encounter expressions involving exponents and fractions when calculating quantities such as velocity and acceleration. In engineering, we often encounter expressions involving exponents and fractions when designing and analyzing complex systems.

Final Thoughts

Simplifying expressions involving exponents and fractions is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. By applying the rules of exponents and fractions carefully, we can simplify complex expressions and arrive at the final simplified expression. With practice and patience, we can become more comfortable with the rules and techniques of simplifying expressions involving exponents and fractions.

Additional Resources

For more information on simplifying expressions involving exponents and fractions, check out the following resources:

• Khan Academy: Exponents and Fractions
• Mathway: Simplifying Expressions Involving Exponents and Fractions
• Wolfram Alpha: Simplifying Expressions Involving Exponents and Fractions

Frequently Asked Questions

Q: What is the rule of exponents? A: The rule of exponents states that when we have a power raised to another power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.

Q: How do I simplify a fraction involving exponents? A: To simplify a fraction involving exponents, make sure to cancel out the common factors in the numerator and denominator.

Q: What is the final simplified expression for the given expression? A: The final simplified expression for the given expression is $-b^2$.

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Introduction

In our previous article, we simplified the expression $\frac{\left(3 a^6 b^3\right)^2}{-9 a^{12} b^4}$ by applying the rules of exponents and fractions. We started by simplifying the numerator and denominator separately, and then we canceled out the common factors to get the final simplified expression. The final simplified expression is $-b^2$. In this article, we will answer some frequently asked questions related to simplifying expressions involving exponents and fractions.

Q&A

Q: What is the rule of exponents?

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

Q: How do I simplify a fraction involving exponents?

A: To simplify a fraction involving exponents, make sure to cancel out the common factors in the numerator and denominator.

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

A: The final simplified expression for the given expression is $-b^2$.

Q: Can I simplify an expression involving exponents and fractions using a calculator?

A: Yes, you can simplify an expression involving exponents and fractions using a calculator. However, it's always a good idea to understand the underlying concepts and rules of exponents and fractions before using a calculator.

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

A: To apply the rule of exponents, follow these steps:

1. Identify the power and the exponent.
2. Multiply the exponents.
3. Simplify the resulting expression.

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

A: A power is the result of raising a number to a certain power, while an exponent is the number that is raised to a certain power.

Q: Can I simplify an expression involving exponents and fractions using a computer algebra system (CAS)?

A: Yes, you can simplify an expression involving exponents and fractions using a CAS. However, it's always a good idea to understand the underlying concepts and rules of exponents and fractions before using a CAS.

Q: How do I simplify an expression involving exponents and fractions with negative exponents?

A: To simplify an expression involving exponents and fractions with negative exponents, follow these steps:

1. Identify the negative exponent.
2. Rewrite the expression with a positive exponent.
3. Simplify the resulting expression.

Q: What is the final simplified expression for the expression $\frac{a^{-3}}{a^2}$?

A: The final simplified expression for the expression $\frac{a^{-3}}{a^2}$ is $a^{-5}$.

Q: Can I simplify an expression involving exponents and fractions with fractional exponents?

A: Yes, you can simplify an expression involving exponents and fractions with fractional exponents. However, it's always a good idea to understand the underlying concepts and rules of exponents and fractions before simplifying the expression.

Q: How do I simplify an expression involving exponents and fractions with mixed exponents?

A: To simplify an expression involving exponents and fractions with mixed exponents, follow these steps:

1. Identify the mixed exponents.
2. Rewrite the expression with a single exponent.
3. Simplify the resulting expression.

Q: What is the final simplified expression for the expression $\frac{a^2 b^3}{a^{-1} b^{-2}}$?

A: The final simplified expression for the expression $\frac{a^2 b^3}{a^{-1} b^{-2}}$ is $a^3 b^5$.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions involving exponents and fractions. We covered topics such as the rule of exponents, simplifying fractions, and simplifying expressions with negative exponents, fractional exponents, and mixed exponents. We also provided examples of how to simplify expressions involving exponents and fractions.

Tips and Tricks

• When simplifying expressions involving exponents and fractions, make sure to apply the rules of exponents and fractions carefully.
• When simplifying fractions, make sure to cancel out the common factors in the numerator and denominator.
• Practice simplifying expressions involving exponents and fractions to become more comfortable with the rules and techniques.

Common Mistakes to Avoid

• When simplifying expressions involving exponents and fractions, make sure to apply the rules of exponents and fractions carefully. For example, $(a^m)^n = a^{m \cdot n}$, not $a^{m + n}$.
• When simplifying fractions, make sure to cancel out the common factors in the numerator and denominator. For example, $\frac{a^m}{a^n} = a^{m-n}$, not $a^m$.

Real-World Applications

Simplifying expressions involving exponents and fractions has many real-world applications. For example, in physics, we often encounter expressions involving exponents and fractions when calculating quantities such as velocity and acceleration. In engineering, we often encounter expressions involving exponents and fractions when designing and analyzing complex systems.

Final Thoughts

Simplifying expressions involving exponents and fractions is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. By applying the rules of exponents and fractions carefully, we can simplify complex expressions and arrive at the final simplified expression. With practice and patience, we can become more comfortable with the rules and techniques of simplifying expressions involving exponents and fractions.

Additional Resources

For more information on simplifying expressions involving exponents and fractions, check out the following resources:

• Khan Academy: Exponents and Fractions
• Mathway: Simplifying Expressions Involving Exponents and Fractions
• Wolfram Alpha: Simplifying Expressions Involving Exponents and Fractions

Frequently Asked Questions

Q: What is the rule of exponents? A: The rule of exponents states that when we have a power raised to another power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.

Q: How do I simplify a fraction involving exponents? A: To simplify a fraction involving exponents, make sure to cancel out the common factors in the numerator and denominator.

Q: What is the final simplified expression for the given expression? A: The final simplified expression for the given expression is $-b^2$.