Select the Value of Each Variable

Oct 13, 2025 by miodeya.com 92 views

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Introduction

When it comes to evaluating expressions with exponents, it's essential to follow a specific order of operations to ensure accuracy. In this article, we'll explore the steps to evaluate the given expression and provide a clear understanding of the process.

The Expression

Consider the expression:

{\frac{4^5(-2)^9}{4^8(-2)^3}\}

This expression involves exponents, multiplication, and division. To evaluate it, we need to follow the order of operations, which is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Step 1: Apply the Quotient of Powers

The quotient of powers rule states that when dividing two powers with the same base, we subtract the exponents. In this case, we have:

{\frac{4^5}{4^8}\}

Using the quotient of powers rule, we can rewrite this as:

{4^{5-8}\}

{4^{-3}\}

Step 3: Evaluate Powers

Now that we have simplified the expression, we can evaluate the powers. The expression becomes:

{\frac{(-2)^9}{(-2)^3(-2)^3}\}

Using the product of powers rule, we can rewrite this as:

{\frac{(-2)^9}{(-2)^6}\}

Step 4: Simplify the Expression

Now that we have simplified the expression, we can evaluate the powers. The expression becomes:

{(-2)^{9-6}\}

{(-2)^3\}

Step 5: Evaluate the Final Expression

Finally, we can evaluate the final expression:

{-8\}

Conclusion

In this article, we explored the steps to evaluate the given expression and provided a clear understanding of the process. By following the order of operations and applying the quotient of powers and product of powers rules, we were able to simplify the expression and evaluate the final result.

Discussion

Step 1: Apply the Quotient of Powers

The quotient of powers rule is a fundamental concept in algebra that allows us to simplify expressions with exponents. In this case, we used the rule to simplify the expression $${\frac{4^5}{48}}$$.

Step 2: Evaluate Powers

After simplifying the expression, we evaluated the powers using the product of powers rule. This rule allows us to multiply powers with the same base.

Step 3: Simplify the Expression

In this step, we simplified the expression by combining the powers. This involved using the product of powers rule to rewrite the expression as $${\frac{(-2)^9}{(-2)6}}$$.

Step 4: Evaluate the Final Expression

Finally, we evaluated the final expression by simplifying the powers. This involved using the quotient of powers rule to rewrite the expression as $${(-2)^{9-6}}$$.

Final Answer

The final answer is $\boxed{-8}$ .

Select the Value of Each Variable

a

{a =\}$ ${$\square\$}

b

{b =\}

Note: The values of a and b are not provided in the original expression, so we cannot determine their values.

Discussion Category

This article falls under the discussion category of mathematics, specifically algebra and exponent rules.

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Introduction

In our previous article, we explored the steps to evaluate the given expression and provided a clear understanding of the process. However, we understand that some readers may still have questions or need further clarification. In this article, we'll address some of the most frequently asked questions related to evaluating expressions with exponents.

Q&A

Q: What is the order of operations when evaluating expressions with exponents?

A: The order of operations when evaluating expressions with exponents is:

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

Q: What is the quotient of powers rule?

A: The quotient of powers rule states that when dividing two powers with the same base, we subtract the exponents. For example, $${\frac{a^m}{an} = a^{m-n}}$$.

Q: What is the product of powers rule?

A: The product of powers rule states that when multiplying two powers with the same base, we add the exponents. For example, $${a^m \cdot a^n = a^{m+n}}$$.

Q: How do I simplify expressions with exponents?

A: To simplify expressions with exponents, follow these steps:

Evaluate any exponential expressions.
Use the quotient of powers rule to simplify any division operations.
Use the product of powers rule to simplify any multiplication operations.
Combine any like terms.

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 raised to a power and then taken as a reciprocal. For example, $${a^3}$$ is different from $${\frac{1}{a^3}}$$.

Q: How do I evaluate expressions with negative exponents?

A: To evaluate expressions with negative exponents, follow these steps:

Rewrite the expression with a positive exponent by taking the reciprocal of the base.
Evaluate the expression as you would with a positive exponent.

Q: What is the rule for evaluating expressions with fractional exponents?

A: The rule for evaluating expressions with fractional exponents is:

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

Q: How do I evaluate expressions with multiple exponents?

A: To evaluate expressions with multiple exponents, follow these steps:

Evaluate any exponential expressions.
Use the quotient of powers rule to simplify any division operations.
Use the product of powers rule to simplify any multiplication operations.
Combine any like terms.

Conclusion

In this article, we addressed some of the most frequently asked questions related to evaluating expressions with exponents. By following the order of operations and applying the quotient of powers and product of powers rules, we can simplify expressions with exponents and evaluate the final result.

Final Answer

The final answer is $\boxed{-8}$ .

Discussion Category

This article falls under the discussion category of mathematics, specifically algebra and exponent rules.

Additional Resources

For further clarification or additional resources, please refer to the following:

Khan Academy: Exponents and Exponential Functions
Mathway: Exponents and Exponential Functions
Wolfram Alpha: Exponents and Exponential Functions

Note: The values of a and b are not provided in the original expression, so we cannot determine their values.