Na Correctly Simplified The Expression $\frac{-4 A^{-6} B^4}{8 A^{-6} B^{-3}}$, Assuming That $a \neq 0$ And $b \neq 0$. Her Simplified Expression Is $-\frac{1}{2} A^0 B^{\square}$.What Should The Exponent Of The

Introduction

Simplifying expressions with exponents can be a challenging task, especially when dealing with negative exponents and fractions. In this article, we will explore how to simplify the expression $\frac{-4 a^{-6} b^4}{8 a^{-6} b^{-3}}$ and determine the correct exponent of the variable $b$.

Understanding Exponents

Before we dive into the simplification process, let's review the basics of exponents. An exponent is a small number that is written to the upper right of a number or variable. It represents the power to which the base is raised. For example, $a^3$ means $a$ raised to the power of 3, or $a \times a \times a$.

Simplifying the Expression

To simplify the expression $\frac{-4 a^{-6} b^4}{8 a^{-6} b^{-3}}$, we need to follow the order of operations (PEMDAS):

1. Parentheses: There are no parentheses in the expression.
2. Exponents: We can simplify the exponents by combining like terms.
3. Multiplication and Division: We can simplify the expression by dividing the numerator and denominator by the common factors.
4. Addition and Subtraction: There are no addition or subtraction operations in the expression.

Let's start by simplifying the exponents:

$$\frac{-4 a^{-6} b^4}{8 a^{-6} b^{-3}} = \frac{-4}{8} \cdot \frac{a^{-6}}{a^{-6}} \cdot \frac{b^4}{b^{-3}}$$

We can simplify the fraction $\frac{-4}{8}$ by dividing both the numerator and denominator by 4:

$$\frac{-4}{8} = \frac{-1}{2}$$

Next, we can simplify the fraction $\frac{a^{-6}}{a^{-6}}$ by canceling out the common factors:

$$\frac{a^{-6}}{a^{-6}} = 1$$

Finally, we can simplify the fraction $\frac{b^4}{b^{-3}}$ by using the rule for dividing exponents with the same base:

$$\frac{b^4}{b^{-3}} = b^{4-(-3)} = b^7$$

Therefore, the simplified expression is:

$$\frac{-4 a^{-6} b^4}{8 a^{-6} b^{-3}} = -\frac{1}{2} b^7$$

Conclusion

In conclusion, the correct exponent of the variable $b$ is 7. Na's simplified expression $-\frac{1}{2} a^0 b^{\square}$ is incorrect, as the exponent of $b$ should be 7, not $\square$.

Tips and Tricks

When simplifying expressions with exponents, remember to:

• Follow the order of operations (PEMDAS)
• Simplify the exponents by combining like terms
• Use the rule for dividing exponents with the same base
• Cancel out common factors

By following these tips and tricks, you can simplify expressions with exponents like a pro!

Common Mistakes

When simplifying expressions with exponents, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not following the order of operations (PEMDAS)
• Not simplifying the exponents by combining like terms
• Not using the rule for dividing exponents with the same base
• Not canceling out common factors

By avoiding these common mistakes, you can ensure that your simplified expression is correct.

Practice Problems

Want to practice simplifying expressions with exponents? Try these practice problems:

1. Simplify the expression $\frac{2 a^3 b^2}{4 a^3 b^2}$
2. Simplify the expression $\frac{-3 a^{-2} b^4}{6 a^{-2} b^4}$
3. Simplify the expression $\frac{5 a^2 b^3}{10 a^2 b^3}$

By practicing these problems, you can improve your skills in simplifying expressions with exponents.

Conclusion

Introduction

In our previous article, we explored how to simplify the expression $\frac{-4 a^{-6} b^4}{8 a^{-6} b^{-3}}$ and determined the correct exponent of the variable $b$. In this article, we will answer some frequently asked questions about simplifying exponents.

Q&A

Q: What is the order of operations for simplifying exponents?

A: The order of operations for simplifying exponents is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify exponents with the same base?

A: To simplify exponents with the same base, you can use the rule:

$$a^m \cdot a^n = a^{m+n}$$

For example, $a^3 \cdot a^4 = a^{3+4} = a^7$.

Q: How do I simplify exponents with different bases?

A: To simplify exponents with different bases, you cannot combine them using the rule above. Instead, you can leave the expression as is or simplify it by canceling out common factors.

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

A: The rule for dividing exponents with the same base is:

$$\frac{a^m}{a^n} = a^{m-n}$$

For example, $\frac{a^3}{a^2} = a^{3-2} = a^1$.

Q: How do I simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, you can use the rule:

$$a^{-m} = \frac{1}{a^m}$$

For example, $a^{-3} = \frac{1}{a^3}$.

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.

Q: Can I simplify an expression with a zero exponent?

A: Yes, an expression with a zero exponent is equal to 1. For example, $a^0 = 1$.

Q: Can I simplify an expression with a fractional exponent?

A: Yes, an expression with a fractional exponent can be simplified using the rule:

$$a^{m/n} = \sqrt[n]{a^m}$$

For example, $a^{2/3} = \sqrt[3]{a^2}$.

Conclusion

In conclusion, simplifying exponents requires careful attention to the order of operations, simplifying exponents, and using the rules for dividing exponents with the same base and negative exponents. By following these tips and tricks, you can simplify expressions with exponents like a pro!

Practice Problems

Want to practice simplifying expressions with exponents? Try these practice problems:

1. Simplify the expression $\frac{2 a^3 b^2}{4 a^3 b^2}$
2. Simplify the expression $\frac{-3 a^{-2} b^4}{6 a^{-2} b^4}$
3. Simplify the expression $\frac{5 a^2 b^3}{10 a^2 b^3}$

By practicing these problems, you can improve your skills in simplifying expressions with exponents.

Common Mistakes

When simplifying expressions with exponents, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not following the order of operations (PEMDAS)
• Not simplifying the exponents by combining like terms
• Not using the rule for dividing exponents with the same base
• Not canceling out common factors

By avoiding these common mistakes, you can ensure that your simplified expression is correct.

Conclusion

In conclusion, simplifying expressions with exponents requires careful attention to the order of operations, simplifying exponents, and using the rules for dividing exponents with the same base and negative exponents. By following these tips and tricks, you can simplify expressions with exponents like a pro!