2.8 Calculate: − 13 ( − 4 + 4 ) ÷ 13 -13(-4+4) \div 13 − 13 ( − 4 + 4 ) ÷ 13 \begin Tabular}{ll} A & -1 \B & 0 \C & -8 \D & $\frac{56}{13}$ \\end{tabular}Show Your Work.---SECTION BQuestion 33.1 Simplify $\sqrt{\frac{\sqrt[3]{-64 +5}{4 2+3 2}}$

Section A: Calculating Expressions

2.8 Calculate: $-13(-4+4) \div 13$

To solve this expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the expression inside the parentheses: $-4+4 = 0$
2. Multiply $-13$ by the result: $-13 \times 0 = 0$
3. Divide the result by $13$: $0 \div 13 = 0$

Therefore, the correct answer is:

0

Explanation

The expression $-13(-4+4) \div 13$ can be solved by following the order of operations. First, we evaluate the expression inside the parentheses, which is $-4+4$. This simplifies to $0$. Then, we multiply $-13$ by the result, which is also $0$. Finally, we divide the result by $13$, which is still $0$.

Section B: Simplifying Expressions

33.1 Simplify: $\sqrt{\frac{\sqrt[3]{-64}+5}{4^2+3^2}}$

To simplify this expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the cube root: $\sqrt[3]{-64} = -4$
2. Add $5$ to the result: $-4 + 5 = 1$
3. Evaluate the expression in the denominator: $4^2 + 3^2 = 16 + 9 = 25$
4. Divide the result from step 2 by the result from step 3: $\frac{1}{25}$
5. Take the square root of the result from step 4: $\sqrt{\frac{1}{25}} = \frac{1}{5}$

Therefore, the correct answer is:

$\frac{1}{5}$

Explanation

The expression $\sqrt{\frac{\sqrt[3]{-64}+5}{4^2+3^2}}$ can be simplified by following the order of operations. First, we evaluate the cube root, which is $\sqrt[3]{-64} = -4$. Then, we add $5$ to the result, which is $-4 + 5 = 1$. Next, we evaluate the expression in the denominator, which is $4^2 + 3^2 = 16 + 9 = 25$. We then divide the result from step 2 by the result from step 3, which is $\frac{1}{25}$. Finally, we take the square root of the result from step 4, which is $\sqrt{\frac{1}{25}} = \frac{1}{5}$.

Conclusion

Solving complex mathematical expressions requires a step-by-step approach and a clear understanding of the order of operations. By following the order of operations (PEMDAS), we can simplify even the most complex expressions and arrive at the correct answer. In this article, we have demonstrated how to solve two complex mathematical expressions, one involving multiplication and division, and the other involving cube roots and square roots. By following the steps outlined in this article, you can become proficient in solving complex mathematical expressions and arrive at the correct answer with confidence.

Key Takeaways

• Follow the order of operations (PEMDAS) when solving complex mathematical expressions.
• Evaluate expressions inside parentheses first.
• Multiply and divide from left to right.
• Add and subtract from left to right.
• Use the correct order of operations to simplify expressions involving cube roots and square roots.

Practice Problems

1. Simplify: $\sqrt{\frac{\sqrt[3]{-27}+3}{2^2+1^2}}$
2. Calculate: $-12(-3+3) \div 12$
3. Simplify: $\sqrt{\frac{\sqrt[3]{-216}+2}{5^2+4^2}}$

Answer Key

1. $\frac{1}{2}$
2. $0$
3. $\frac{1}{5}$
   Frequently Asked Questions: Solving Complex Mathematical 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. PEMDAS stands for:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next (e.g., 2^3).
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 evaluate expressions inside parentheses?

A: To evaluate expressions inside parentheses, simply follow the order of operations (PEMDAS) within the parentheses. For example, if we have the expression $2(3+4)$, we would first evaluate the expression inside the parentheses: $3+4 = 7$. Then, we would multiply 2 by the result: $2 \times 7 = 14$.

Q: What is the difference between multiplication and division?

A: Multiplication and division are both operations that involve combining numbers. However, multiplication involves adding a number a certain number of times, while division involves sharing a number into equal groups. For example, $3 \times 4 = 12$ (adding 3 together 4 times), while $12 \div 3 = 4$ (sharing 12 into 3 equal groups).

Q: How do I simplify expressions involving cube roots and square roots?

A: To simplify expressions involving cube roots and square roots, we need to follow the order of operations (PEMDAS). First, we evaluate any expressions inside parentheses. Then, we evaluate any exponential expressions (e.g., cube roots and square roots). Finally, we evaluate any multiplication and division operations from left to right.

Q: What is the difference between a cube root and a square root?

A: A cube root is a number that, when multiplied by itself twice, gives the original number. For example, $\sqrt[3]{8} = 2$ because $2 \times 2 \times 2 = 8$. A square root is a number that, when multiplied by itself, gives the original number. For example, $\sqrt{16} = 4$ because $4 \times 4 = 16$.

Q: How do I simplify expressions involving fractions?

A: To simplify expressions involving fractions, we need to follow the order of operations (PEMDAS). First, we evaluate any expressions inside parentheses. Then, we evaluate any exponential expressions (e.g., cube roots and square roots). Finally, we evaluate any multiplication and division operations from left to right.

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

A: The numerator is the top number in a fraction, while the denominator is the bottom number. For example, in the fraction $\frac{3}{4}$, 3 is the numerator and 4 is the denominator.

Q: How do I simplify expressions involving decimals?

A: To simplify expressions involving decimals, we need to follow the order of operations (PEMDAS). First, we evaluate any expressions inside parentheses. Then, we evaluate any exponential expressions (e.g., cube roots and square roots). Finally, we evaluate any multiplication and division operations from left to right.

Q: What is the difference between a decimal and a fraction?

A: A decimal is a way of writing a fraction with a denominator of 10 or a power of 10. For example, the fraction $\frac{3}{4}$ can be written as the decimal 0.75.

Conclusion

Solving complex mathematical expressions requires a step-by-step approach and a clear understanding of the order of operations (PEMDAS). By following the order of operations and simplifying expressions involving cube roots, square roots, fractions, and decimals, we can arrive at the correct answer with confidence. In this article, we have answered some of the most frequently asked questions about solving complex mathematical expressions. By practicing and applying these concepts, you can become proficient in solving complex mathematical expressions and arrive at the correct answer with ease.

Practice Problems

1. Simplify: $\sqrt{\frac{\sqrt[3]{-27}+3}{2^2+1^2}}$
2. Calculate: $-12(-3+3) \div 12$
3. Simplify: $\sqrt{\frac{\sqrt[3]{-216}+2}{5^2+4^2}}$
4. Simplify: $\frac{1}{2} \times \frac{3}{4}$
5. Simplify: $0.5 \times 0.75$

Answer Key

1. $\frac{1}{2}$
2. $0$
3. $\frac{1}{5}$
4. $\frac{3}{8}$
5. $0.375$