10. Simplify The Expression: $\[ 15 - \left(9 + \frac{24}{16 + 2}\right) \\]11. Simplify The Expression: $\[ 4\left(\frac{7 + 14}{3} + 6\right) \\]

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Introduction

Mathematical expressions can be complex and challenging to simplify, especially when they involve fractions, parentheses, and multiple operations. In this article, we will focus on simplifying two complex mathematical expressions using basic algebraic rules and techniques.

Expression 1: Simplifying the Expression 15 - (9 + 24/(16 + 2))

The first expression we will simplify is 15 - (9 + 24/(16 + 2)). To simplify this expression, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expression inside the parentheses: 16 + 2 = 18
  2. Evaluate the fraction: 24/18 = 4/3
  3. Simplify the expression inside the parentheses: 9 + 4/3 = 9 + 1.33 = 10.33
  4. Subtract 10.33 from 15: 15 - 10.33 = 4.67

Therefore, the simplified expression is 4.67.

Step-by-Step Solution

Step 1: Evaluate the Expression Inside the Parentheses

The expression inside the parentheses is 16 + 2. To evaluate this expression, we simply add 16 and 2:

16 + 2 = 18

Step 2: Evaluate the Fraction

The fraction is 24/18. To evaluate this fraction, we simply divide 24 by 18:

24/18 = 4/3

Step 3: Simplify the Expression Inside the Parentheses

The expression inside the parentheses is 9 + 4/3. To simplify this expression, we can convert the fraction to a decimal and then add it to 9:

9 + 4/3 = 9 + 1.33 = 10.33

Step 4: Subtract 10.33 from 15

Finally, we subtract 10.33 from 15:

15 - 10.33 = 4.67

Expression 2: Simplifying the Expression 4((7 + 14)/3 + 6)

The second expression we will simplify is 4((7 + 14)/3 + 6). To simplify this expression, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expression inside the parentheses: 7 + 14 = 21
  2. Divide 21 by 3: 21/3 = 7
  3. Add 6 to 7: 7 + 6 = 13
  4. Multiply 4 by 13: 4 * 13 = 52

Therefore, the simplified expression is 52.

Step-by-Step Solution

Step 1: Evaluate the Expression Inside the Parentheses

The expression inside the parentheses is 7 + 14. To evaluate this expression, we simply add 7 and 14:

7 + 14 = 21

Step 2: Divide 21 by 3

Next, we divide 21 by 3:

21/3 = 7

Step 3: Add 6 to 7

Now, we add 6 to 7:

7 + 6 = 13

Step 4: Multiply 4 by 13

Finally, we multiply 4 by 13:

4 * 13 = 52

Conclusion

Introduction

In our previous article, we explored two complex mathematical expressions and simplified them using basic algebraic rules and techniques. In this article, we will answer some frequently asked questions (FAQs) related to simplifying 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. The acronym 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 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 simplify an expression with multiple parentheses?

A: To simplify an expression with multiple parentheses, follow these steps:

  1. Evaluate the innermost parentheses first.
  2. Work your way outwards, evaluating each set of parentheses in turn.
  3. Once you have simplified all the parentheses, you can perform any remaining operations.

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

A: A fraction is a way of expressing a part of a whole as a ratio of two numbers. For example, 1/2 is a fraction. A decimal is a way of expressing a fraction as a number with a point (.) separating the whole number part from the fractional part. For example, 0.5 is a decimal.

Q: How do I convert a fraction to a decimal?

A: To convert a fraction to a decimal, simply divide the numerator (the top number) by the denominator (the bottom number). For example, to convert 1/2 to a decimal, divide 1 by 2:

1 ÷ 2 = 0.5

Q: What is the difference between an exponent and a power?

A: An exponent is a small number that is raised to a power. For example, 2^3 is an exponent. A power is the result of raising a number to a power. For example, 2^3 = 8 is a power.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, follow these steps:

  1. Evaluate any exponential expressions first.
  2. Simplify any remaining expressions using the order of operations (PEMDAS).

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change. For example, x is a variable. A constant is a value that does not change. For example, 5 is a constant.

Q: How do I simplify an expression with variables?

A: To simplify an expression with variables, follow these steps:

  1. Evaluate any exponential expressions first.
  2. Simplify any remaining expressions using the order of operations (PEMDAS).
  3. Combine like terms (terms with the same variable).

Conclusion

Simplifying complex mathematical expressions requires careful attention to the order of operations and the use of basic algebraic rules and techniques. By following the steps outlined in this article, you can simplify even the most complex expressions and arrive at the correct solution. Remember to always 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 simplifying complex mathematical expressions and solving a wide range of mathematical problems.