Which Expression Is Equal To $-3 B^4\left(6 B^{-8}\right$\]? Assume $b \neq 0$.A. $-18 B^{-4}$B. $18 B^{-4}$C. $-18 B^4$D. $18 B^4$

Understanding Exponential Notation

Exponential notation is a shorthand way of writing repeated multiplication. For example, $b^4$ means $b \times b \times b \times b$. When we see a negative exponent, it means we are dealing with a fraction. For instance, $b^{-4}$ means $\frac{1}{b^4}$.

Simplifying the Given Expression

The given expression is $-3 b^4\left(6 b^{-8}\right)$. To simplify this expression, we need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate the expression inside the parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication: Multiply the numbers together.
4. Division: Divide the numbers.

Step 1: Evaluate the Expression Inside the Parentheses

The expression inside the parentheses is $6 b^{-8}$. We can rewrite this as $\frac{6}{b^8}$.

Step 2: Multiply the Numbers Together

Now we can multiply the numbers together:

$-3 b^4 \times \frac{6}{b^8}$

Step 3: Simplify the Expression

To simplify the expression, we can multiply the numbers together and combine the exponents:

$-3 \times 6 \times b^4 \times b^{-8}$

$-18 \times b^{4-8}$

$-18 \times b^{-4}$

The Final Answer

The final answer is $-18 b^{-4}$.

Why is the Answer Correct?

The answer is correct because we followed the order of operations and simplified the expression correctly. We evaluated the expression inside the parentheses, multiplied the numbers together, and combined the exponents.

Comparison with Other Options

Let's compare our answer with the other options:

• A. $-18 b^{-4}$: This is the same as our answer.
• B. $18 b^{-4}$: This is incorrect because the negative sign is missing.
• C. $-18 b^4$: This is incorrect because the exponent is incorrect.
• D. $18 b^4$: This is incorrect because the negative sign is missing and the exponent is incorrect.

Conclusion

In conclusion, the correct answer is $-18 b^{-4}$. We simplified the expression correctly by following the order of operations and combining the exponents.

Common Mistakes to Avoid

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

• Not following the order of operations (PEMDAS)
• Not combining exponents correctly
• Not simplifying fractions correctly

Tips for Simplifying Exponential Expressions

Here are some tips for simplifying exponential expressions:

• Follow the order of operations (PEMDAS)
• Combine exponents correctly
• Simplify fractions correctly
• Use negative exponents to represent fractions

Practice Problems

Here are some practice problems to help you practice simplifying exponential expressions:

1. Simplify the expression: $2 b^3 \left(4 b^{-2}\right)$
2. Simplify the expression: $-5 a^2 \left(3 a^{-4}\right)$
3. Simplify the expression: $6 c^4 \left(2 c^{-3}\right)$

Answer Key

Here are the answers to the practice problems:

1. $-8 b^{-1}$
2. $-15 a^{-2}$
3. $12 c^{1}$
   Frequently Asked Questions: Simplifying Exponential Expressions ====================================================================

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication: Multiply numbers together.
4. Division: Divide numbers.

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

A: To simplify an expression with a negative exponent, we can rewrite it as a fraction. For example, $b^{-4}$ can be rewritten as $\frac{1}{b^4}$.

Q: What is the difference between $b^4$ and $b^{-4}$?

A: $b^4$ means $b$ multiplied by itself 4 times, while $b^{-4}$ means $\frac{1}{b^4}$. In other words, $b^{-4}$ is the reciprocal of $b^4$.

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

A: To simplify an expression with multiple exponents, we can combine the exponents by adding or subtracting them. For example, $b^4 \times b^{-2}$ can be simplified to $b^{4-2} = b^2$.

Q: What is the rule for multiplying exponential expressions?

A: When multiplying exponential expressions, we add the exponents. For example, $b^4 \times b^2 = b^{4+2} = b^6$.

Q: What is the rule for dividing exponential expressions?

A: When dividing exponential expressions, we subtract the exponents. For example, $\frac{b^4}{b^2} = b^{4-2} = b^2$.

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

A: To simplify an expression with a fraction and an exponent, we can multiply the fraction by the exponent. For example, $\frac{1}{b^4} \times b^2 = \frac{b^2}{b^4} = b^{-2}$.

Q: What is the difference between $b^4$ and $b^{4+2}$?

A: $b^4$ means $b$ multiplied by itself 4 times, while $b^{4+2}$ means $b$ multiplied by itself 6 times. In other words, $b^{4+2}$ is the same as $b^6$.

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

A: To simplify an expression with a negative exponent and a fraction, we can rewrite the fraction as a negative exponent. For example, $\frac{1}{b^4} = b^{-4}$.

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

A: The rule for simplifying exponential expressions with variables is the same as for numerical exponents. We can combine exponents by adding or subtracting them, and we can multiply or divide exponential expressions by adding or subtracting the exponents.

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

A: To simplify an expression with multiple variables and exponents, we can combine the exponents by adding or subtracting them, and we can multiply or divide exponential expressions by adding or subtracting the exponents.

Q: What is the difference between $b^4$ and $b^{4+2}$ when $b$ is a variable?

A: $b^4$ means $b$ multiplied by itself 4 times, while $b^{4+2}$ means $b$ multiplied by itself 6 times. In other words, $b^{4+2}$ is the same as $b^6$.

Conclusion

In conclusion, simplifying exponential expressions requires a clear understanding of the rules for combining exponents and multiplying or dividing exponential expressions. By following the order of operations (PEMDAS) and combining exponents correctly, we can simplify even the most complex exponential expressions.