Simplify The Expression:${ \frac{30 A^8 B^2 C^{-6}}{5 A^{-9} B^{-7}} }$Write Your Answer Using Only Positive Exponents.

$$

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it is essential to understand the rules of exponents to simplify complex expressions. In this article, we will focus on simplifying the given expression using the rules of exponents. We will use the properties of exponents to rewrite the expression with only positive exponents.

Understanding the Rules of Exponents

Before we dive into simplifying the expression, let's review the rules of exponents. The rules of exponents state that when we multiply two numbers with the same base, we add their exponents. For example, $a^m \cdot a^n = a^{m+n}$. Similarly, when we divide two numbers with the same base, we subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Simplifying the Expression

Now that we have reviewed the rules of exponents, let's simplify the given expression. The expression is $\frac{30 a^8 b^2 c^{-6}}{5 a^{-9} b^{-7}}$. To simplify this expression, we need to apply the rules of exponents.

First, let's simplify the coefficients. The coefficient of the numerator is 30, and the coefficient of the denominator is 5. We can simplify the coefficients by dividing 30 by 5, which gives us 6.

Next, let's simplify the exponents. We can start by simplifying the exponents of the variables $a$ and $b$. Using the rule of exponents, we can rewrite the expression as:

$$\frac{6 a^{8-(-9)} b^{2-(-7)}}{c^{-6}}$$

Now, let's simplify the exponents of $a$ and $b$. Using the rule of exponents, we can rewrite the expression as:

$$\frac{6 a^{17} b^9}{c^{-6}}$$

Applying the Quotient Rule

Now that we have simplified the exponents of $a$ and $b$, let's apply the quotient rule to simplify the expression further. The quotient rule states that when we divide two numbers with the same base, we subtract their exponents. In this case, we can rewrite the expression as:

$$6 a^{17} b^9 c^6$$

Conclusion

In this article, we simplified the given expression using the rules of exponents. We started by simplifying the coefficients and then applied the rules of exponents to simplify the exponents of the variables $a$ and $b$. Finally, we applied the quotient rule to simplify the expression further. The simplified expression is $6 a^{17} b^9 c^6$.

Final Answer

The final answer is $\boxed{6 a^{17} b^9 c^6}$.

Discussion

Simplifying algebraic expressions is a crucial skill in mathematics, and it is essential to understand the rules of exponents to simplify complex expressions. In this article, we focused on simplifying the given expression using the rules of exponents. We applied the rules of exponents to simplify the exponents of the variables $a$ and $b$ and then applied the quotient rule to simplify the expression further. The simplified expression is $6 a^{17} b^9 c^6$.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

• Always start by simplifying the coefficients.
• Use the rules of exponents to simplify the exponents of the variables.
• Apply the quotient rule to simplify the expression further.
• Make sure to check your work by plugging in values for the variables.

Common Mistakes

Here are some common mistakes to avoid when simplifying algebraic expressions:

• Not simplifying the coefficients.
• Not applying the rules of exponents.
• Not applying the quotient rule.
• Not checking your work.

Real-World Applications

Simplifying algebraic expressions has many real-world applications. For example, in physics, we use algebraic expressions to describe the motion of objects. In engineering, we use algebraic expressions to design and optimize systems. In economics, we use algebraic expressions to model and analyze economic systems.

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics, and it is essential to understand the rules of exponents to simplify complex expressions. In this article, we focused on simplifying the given expression using the rules of exponents. We applied the rules of exponents to simplify the exponents of the variables $a$ and $b$ and then applied the quotient rule to simplify the expression further. The simplified expression is $6 a^{17} b^9 c^6$.

$$

Introduction

In our previous article, we simplified the given expression using the rules of exponents. We applied the rules of exponents to simplify the exponents of the variables $a$ and $b$ and then applied the quotient rule to simplify the expression further. The simplified expression is $6 a^{17} b^9 c^6$. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What are the rules of exponents?

A: The rules of exponents state that when we multiply two numbers with the same base, we add their exponents. For example, $a^m \cdot a^n = a^{m+n}$. Similarly, when we divide two numbers with the same base, we subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Q: How do I simplify the coefficients in an algebraic expression?

A: To simplify the coefficients in an algebraic expression, you need to divide the numerator by the denominator. For example, in the expression $\frac{30 a^8 b^2 c^{-6}}{5 a^{-9} b^{-7}}$, you can simplify the coefficients by dividing 30 by 5, which gives you 6.

Q: How do I simplify the exponents in an algebraic expression?

A: To simplify the exponents in an algebraic expression, you need to apply the rules of exponents. For example, in the expression $\frac{30 a^8 b^2 c^{-6}}{5 a^{-9} b^{-7}}$, you can simplify the exponents by adding the exponents of the variables $a$ and $b$. This gives you $a^{8-(-9)} b^{2-(-7)}$.

Q: What is the quotient rule in algebra?

A: The quotient rule in algebra states that when we divide two numbers with the same base, we subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Q: How do I apply the quotient rule in an algebraic expression?

A: To apply the quotient rule in an algebraic expression, you need to subtract the exponents of the variables. For example, in the expression $\frac{30 a^8 b^2 c^{-6}}{5 a^{-9} b^{-7}}$, you can apply the quotient rule by subtracting the exponents of the variables $a$ and $b$. This gives you $a^{17} b^9 c^6$.

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

A: Some common mistakes to avoid when simplifying algebraic expressions include not simplifying the coefficients, not applying the rules of exponents, not applying the quotient rule, and not checking your work.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has many real-world applications. For example, in physics, we use algebraic expressions to describe the motion of objects. In engineering, we use algebraic expressions to design and optimize systems. In economics, we use algebraic expressions to model and analyze economic systems.

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics, and it is essential to understand the rules of exponents to simplify complex expressions. In this article, we answered some frequently asked questions about simplifying algebraic expressions. We hope that this article has been helpful in clarifying any doubts you may have had about simplifying algebraic expressions.

Final Answer

The final answer is $\boxed{6 a^{17} b^9 c^6}$.

Discussion

Simplifying algebraic expressions is a crucial skill in mathematics, and it is essential to understand the rules of exponents to simplify complex expressions. In this article, we focused on simplifying the given expression using the rules of exponents. We applied the rules of exponents to simplify the exponents of the variables $a$ and $b$ and then applied the quotient rule to simplify the expression further. The simplified expression is $6 a^{17} b^9 c^6$.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

• Always start by simplifying the coefficients.
• Use the rules of exponents to simplify the exponents of the variables.
• Apply the quotient rule to simplify the expression further.
• Make sure to check your work by plugging in values for the variables.

Common Mistakes

Here are some common mistakes to avoid when simplifying algebraic expressions:

• Not simplifying the coefficients.
• Not applying the rules of exponents.
• Not applying the quotient rule.
• Not checking your work.

Real-World Applications

Simplifying algebraic expressions has many real-world applications. For example, in physics, we use algebraic expressions to describe the motion of objects. In engineering, we use algebraic expressions to design and optimize systems. In economics, we use algebraic expressions to model and analyze economic systems.