{ (6+5)+6^2 \times 7$}$2) ${$10 \times 10(9+12 \div 2)$}$3) ${$7+30 \div 3(7+10-3)$}$4) { \left(8+3^2\right) \times 6+2$}$5) ${$3+6(50 \div 5+11-4)$}$6) ${$10+2^2(30 \div 3+9)$}$7)

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Introduction

Mathematical expressions can be complex and challenging to solve, especially when they involve multiple operations and parentheses. In this article, we will explore seven complex mathematical expressions and provide a step-by-step guide on how to solve them. We will use the order of operations (PEMDAS) to simplify each expression and arrive at the final answer.

Expression 1: (6+5)+62×7(6+5)+6^2 \times 7

Step 1: Evaluate the expression inside the parentheses

The expression inside the parentheses is 6+56+5, which equals 1111.

Step 2: Evaluate the exponent

The exponent is 626^2, which equals 3636.

Step 3: Multiply the result of the exponent by 7

36×7=25236 \times 7 = 252

Step 4: Add the result of the exponent to the result of the expression inside the parentheses

11+252=26311 + 252 = 263

Final Answer

The final answer to expression 1 is 263\boxed{263}.

Expression 2: 10×10(9+12÷2)10 \times 10(9+12 \div 2)

Step 1: Evaluate the expression inside the parentheses

The expression inside the parentheses is 9+12÷29+12 \div 2, which equals 9+6=159+6 = 15.

Step 2: Multiply 10 by the result of the expression inside the parentheses

10×15=15010 \times 15 = 150

Step 3: Multiply the result of the multiplication by 10

150×10=1500150 \times 10 = 1500

Final Answer

The final answer to expression 2 is 1500\boxed{1500}.

Expression 3: 7+30÷3(7+10−3)7+30 \div 3(7+10-3)

Step 1: Evaluate the expression inside the parentheses

The expression inside the parentheses is 7+10−37+10-3, which equals 1414.

Step 2: Divide 30 by 3

30÷3=1030 \div 3 = 10

Step 3: Multiply the result of the division by the result of the expression inside the parentheses

10×14=14010 \times 14 = 140

Step 4: Add 7 to the result of the multiplication

7+140=1477 + 140 = 147

Final Answer

The final answer to expression 3 is 147\boxed{147}.

Expression 4: (8+32)×6+2\left(8+3^2\right) \times 6+2

Step 1: Evaluate the exponent

The exponent is 323^2, which equals 99.

Step 2: Add 8 to the result of the exponent

8+9=178 + 9 = 17

Step 3: Multiply the result of the addition by 6

17×6=10217 \times 6 = 102

Step 4: Add 2 to the result of the multiplication

102+2=104102 + 2 = 104

Final Answer

The final answer to expression 4 is 104\boxed{104}.

Expression 5: 3+6(50÷5+11−4)3+6(50 \div 5+11-4)

Step 1: Divide 50 by 5

50÷5=1050 \div 5 = 10

Step 2: Add 11 to the result of the division

10+11=2110 + 11 = 21

Step 3: Subtract 4 from the result of the addition

21−4=1721 - 4 = 17

Step 4: Multiply 6 by the result of the subtraction

6×17=1026 \times 17 = 102

Step 5: Add 3 to the result of the multiplication

3+102=1053 + 102 = 105

Final Answer

The final answer to expression 5 is 105\boxed{105}.

Expression 6: 10+22(30÷3+9)10+2^2(30 \div 3+9)

Step 1: Divide 30 by 3

30÷3=1030 \div 3 = 10

Step 2: Add 9 to the result of the division

10+9=1910 + 9 = 19

Step 3: Evaluate the exponent

The exponent is 222^2, which equals 44.

Step 4: Multiply the result of the exponent by the result of the addition

4×19=764 \times 19 = 76

Step 5: Add 10 to the result of the multiplication

10+76=8610 + 76 = 86

Final Answer

The final answer to expression 6 is 86\boxed{86}.

Conclusion

Solving complex mathematical expressions requires a step-by-step approach and a clear understanding of the order of operations. By following the steps outlined in this article, you can simplify even the most complex expressions and arrive at the final answer. Remember to evaluate expressions inside parentheses first, followed by exponents, multiplication and division, and finally addition and subtraction. With practice and patience, you will become proficient in solving complex mathematical expressions and be able to tackle even the most challenging problems.

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Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when there are multiple operations in an expression. The order of operations is:

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

A: To evaluate expressions inside parentheses, simply follow the order of operations and evaluate the expression inside the parentheses first. For example, if you have the expression (2+3), you would first evaluate the expression inside the parentheses, which would be 2+3=5.

Q: What is the difference between multiplication and division?

A: Multiplication and division are both operations that involve numbers, but they have different effects on the numbers. Multiplication involves adding a number a certain number of times, while division involves finding the number that, when multiplied by another number, gives a certain result.

Q: How do I evaluate exponents?

A: To evaluate exponents, simply raise the base number to the power of the exponent. For example, if you have the expression 2^3, you would raise 2 to the power of 3, which would be 222=8.

Q: What is the difference between addition and subtraction?

A: Addition and subtraction are both operations that involve numbers, but they have different effects on the numbers. Addition involves combining two or more numbers to get a total, while subtraction involves finding the difference between two numbers.

Q: How do I evaluate expressions with multiple operations?

A: To evaluate expressions with multiple operations, simply follow the order of operations and evaluate the expression from left to right. For example, if you have the expression 2+34, you would first evaluate the multiplication operation (34=12), and then evaluate the addition operation (2+12=14).

Q: What are some common mistakes to avoid when solving complex mathematical expressions?

A: Some common mistakes to avoid when solving complex mathematical expressions include:

  • Not following the order of operations
  • Not evaluating expressions inside parentheses first
  • Not evaluating exponents correctly
  • Not performing multiplication and division operations from left to right
  • Not performing addition and subtraction operations from left to right

Q: How can I practice solving complex mathematical expressions?

A: There are many ways to practice solving complex mathematical expressions, including:

  • Working through practice problems in a textbook or online resource
  • Using online tools or apps to generate random expressions to solve
  • Creating your own expressions to solve and challenging yourself to solve them
  • Joining a study group or working with a tutor to practice solving expressions together

Conclusion

Solving complex mathematical expressions can be challenging, but with practice and patience, you can become proficient in solving even the most difficult expressions. By following the order of operations and avoiding common mistakes, you can ensure that you are solving expressions correctly and accurately. Remember to practice regularly and challenge yourself to solve increasingly complex expressions. With time and effort, you will become a master of solving complex mathematical expressions.