Question 11When Simplifying ${ \left(64-\left(4 3\right) {\frac{1}{2}}\right) \div\left(\frac{1}{\sqrt{81}}+1\right) }$the Answer, Rounded To Two Decimals, Is:Select One:A. 5.60B. 56.80C. 69.53D. 50.40

Understanding the Problem

When simplifying mathematical expressions, it's essential to follow the correct order of operations to avoid confusion and ensure accuracy. In this article, we'll break down the given expression and simplify it step by step.

The Given Expression

The expression we need to simplify is:

$$\left(64-\left(4^3\right)^{\frac{1}{2}}\right) \div\left(\frac{1}{\sqrt{81}}+1\right)$$

Step 1: Evaluate the Exponent

The first step is to evaluate the exponent $4^3$. This means multiplying 4 by itself three times:

$$4^3 = 4 \times 4 \times 4 = 64$$

Step 2: Simplify the Square Root

Next, we need to simplify the square root of 81:

$$\sqrt{81} = 9$$

Step 3: Simplify the Fraction

Now, we can simplify the fraction $\frac{1}{\sqrt{81}}$:

$$\frac{1}{\sqrt{81}} = \frac{1}{9}$$

Step 4: Simplify the Expression Inside the Parentheses

We can now simplify the expression inside the parentheses:

$$\left(4^3\right)^{\frac{1}{2}} = 64^{\frac{1}{2}} = \sqrt{64} = 8$$

Step 5: Simplify the Expression

Now, we can simplify the entire expression:

$$\left(64-8\right) \div\left(\frac{1}{9}+1\right) = 56 \div\left(\frac{10}{9}\right)$$

Step 6: Simplify the Division

To simplify the division, we can multiply the numerator by the reciprocal of the denominator:

$$56 \div\left(\frac{10}{9}\right) = 56 \times \frac{9}{10} = 50.4$$

Conclusion

After simplifying the expression step by step, we get the final answer:

$$50.4$$

Rounding to Two Decimals

The answer is rounded to two decimals, which is:

$$50.40$$

Answer

The correct answer is:

D. 50.40

Key Takeaways

• When simplifying expressions, it's essential to follow the correct order of operations.
• Evaluate exponents and simplify square roots before simplifying fractions.
• Use the reciprocal of the denominator to simplify division.
• Round the final answer to the required number of decimals.
  Simplifying Expressions: A Q&A Guide =====================================

Frequently Asked Questions

In this article, we'll address some common questions related to simplifying expressions. Whether you're a student or a teacher, these questions and answers will help you better understand the process of simplifying expressions.

Q: What is the correct order of operations?

A: The correct order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate 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 expressions with exponents?

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

1. Evaluate the exponent by multiplying the base by itself the number of times indicated by the exponent.
2. Simplify any expressions inside the parentheses.
3. Combine like terms.

Q: How do I simplify expressions with square roots?

A: To simplify expressions with square roots, follow these steps:

1. Simplify any expressions inside the parentheses.
2. Evaluate the square root by finding the number that, when multiplied by itself, gives the original number.
3. Combine like terms.

Q: How do I simplify expressions with fractions?

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

1. Simplify any expressions inside the parentheses.
2. Evaluate the fraction by dividing the numerator by the denominator.
3. Combine like terms.

Q: What is the difference between a numerator and a denominator?

A: The numerator is the top number in a fraction, and the denominator is the bottom number. For example, in the fraction $\frac{1}{2}$, the numerator is 1 and the denominator is 2.

Q: How do I simplify expressions with multiple operations?

A: To simplify expressions with multiple operations, follow these steps:

1. Evaluate any expressions inside parentheses.
2. Evaluate any exponential expressions.
3. Evaluate any multiplication and division operations from left to right.
4. Finally, evaluate any addition and subtraction operations from left to right.

Q: What is the importance of simplifying expressions?

A: Simplifying expressions is important because it helps to:

• Reduce the complexity of an expression
• Make it easier to evaluate
• Identify any errors or inconsistencies
• Improve understanding and communication of mathematical concepts

Q: How do I know when to round my answer?

A: You should round your answer when:

• The problem specifies a certain number of decimal places.
• You are working with a large or small number that is difficult to work with in its exact form.
• You need to present your answer in a specific format.

Conclusion

Simplifying expressions is an essential skill in mathematics. By following the correct order of operations and simplifying expressions step by step, you can ensure accuracy and clarity in your work. Whether you're a student or a teacher, these questions and answers will help you better understand the process of simplifying expressions.