Simplify The Following Expressions:1. \[$ 4.2 \cdot 2 - 3 - (-2)(5) - (-4)^2 \$\]2. $\[ 4.2 \cdot 3 \frac{3^3 - (-\sqrt{4})^2 + \sqrt[3]{-64}}{-4^2 \times 1^3 + 17} \\]

Feb 26, 2025 by ADMIN 169 views

Simplifying Complex Mathematical Expressions

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In mathematics, simplifying complex expressions is an essential skill that helps in solving problems efficiently. It involves breaking down the expression into smaller parts, applying the order of operations, and simplifying each part before combining them. In this article, we will simplify two complex mathematical expressions using the order of operations and basic algebraic rules.

Expression 1: Simplifying a Numerical Expression

The first expression is: $4.2 \cdot 2 - 3 - (-2)(5) - (-4)^2$

Step 1: Multiply 4.2 and 2

To simplify the expression, we start by multiplying 4.2 and 2.

$4.2 \cdot 2 = 8.4$

Step 2: Multiply -2 and 5

Next, we multiply -2 and 5.

$(-2)(5) = -10$

Step 3: Calculate the Square of -4

Now, we calculate the square of -4.

$(-4)^2 = 16$

Step 4: Substitute the Values

Substitute the calculated values back into the original expression.

$8.4 - 3 - (-10) - 16$

Step 5: Simplify the Expression

Now, we simplify the expression by combining like terms.

$8.4 - 3 + 10 - 16 = -0.6$

The final simplified expression is: $-0.6$

Expression 2: Simplifying a Complex Algebraic Expression

The second expression is: $4.2 \cdot 3 \frac{3^3 - (-\sqrt{4})^2 + \sqrt[3]{-64}}{-4^2 \times 1^3 + 17}$

Step 1: Evaluate the Exponents

To simplify the expression, we start by evaluating the exponents.

$3^3 = 27$

$(-\sqrt{4})^2 = 4$

$\sqrt[3]{-64} = -4$

Step 2: Substitute the Values

Substitute the calculated values back into the expression.

$4.2 \cdot 3 \frac{27 - 4 - 4}{-4^2 \times 1^3 + 17}$

Step 3: Simplify the Numerator

Now, we simplify the numerator by combining like terms.

$27 - 4 - 4 = 19$

Step 4: Simplify the Denominator

Next, we simplify the denominator.

$-4^2 \times 1^3 + 17 = -16 + 17 = 1$

Step 5: Substitute the Values

Substitute the calculated values back into the expression.

$4.2 \cdot 3 \frac{19}{1}$

Step 6: Simplify the Expression

Now, we simplify the expression by multiplying 4.2 and 3.

$4.2 \cdot 3 = 12.6$

$12.6 \cdot 19 = 239.4$

The final simplified expression is: $239.4$

Conclusion

Simplifying complex mathematical expressions requires a thorough understanding of the order of operations and basic algebraic rules. By breaking down the expression into smaller parts, applying the order of operations, and simplifying each part before combining them, we can simplify even the most complex expressions. In this article, we simplified two complex mathematical expressions using the order of operations and basic algebraic rules. The final simplified expressions are: $-0.6$ and $239.4$ .

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In the previous article, we simplified two complex mathematical expressions using the order of operations and basic algebraic rules. In this article, we will answer some frequently asked questions related to simplifying complex mathematical expressions.

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 simplifying an expression. The order of operations 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: How do I simplify a complex expression with multiple operations?

A: To simplify a complex expression with multiple operations, follow these steps:

Identify the operations in the expression and group them into categories (e.g., multiplication, division, addition, and subtraction).
Evaluate any operations inside parentheses first.
Evaluate any exponential expressions next.
Evaluate any multiplication and division operations from left to right.
Finally, evaluate any addition and subtraction operations from left to right.

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

A: A variable is a symbol that represents a value that can change, while a constant is a value that does not change. For example, in the expression $x + 5$ , $x$ is a variable, while $5$ is a constant.

Q: How do I simplify an expression with a variable?

A: To simplify an expression with a variable, follow these steps:

Identify the variable and any constants in the expression.
Evaluate any operations inside parentheses first.
Evaluate any exponential expressions next.
Evaluate any multiplication and division operations from left to right.
Finally, evaluate any addition and subtraction operations from left to right.

Q: What is the difference between a positive and negative exponent?

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the reciprocal of the base is raised to a power. For example, in the expression $2^3$ , the exponent is positive, while in the expression $2^{-3}$ , the exponent is negative.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, follow these steps:

Rewrite the expression with a positive exponent by taking the reciprocal of the base.
Evaluate any operations inside parentheses first.
Evaluate any exponential expressions next.
Evaluate any multiplication and division operations from left to right.
Finally, evaluate any addition and subtraction operations from left to right.

Q: What is the difference between a rational and irrational number?

A: A rational number is a number that can be expressed as a fraction, while an irrational number is a number that cannot be expressed as a fraction. For example, in the expression $\frac{1}{2}$ , the number is rational, while in the expression $\sqrt{2}$ , the number is irrational.

Q: How do I simplify an expression with a rational or irrational number?

A: To simplify an expression with a rational or irrational number, follow these steps:

Identify the rational or irrational number and any constants in the expression.
Evaluate any operations inside parentheses first.
Evaluate any exponential expressions next.
Evaluate any multiplication and division operations from left to right.
Finally, evaluate any addition and subtraction operations from left to right.

Conclusion

Simplifying complex mathematical expressions requires a thorough understanding of the order of operations and basic algebraic rules. By following the steps outlined in this article, you can simplify even the most complex expressions. Remember to identify the operations in the expression, group them into categories, and evaluate them in the correct order. With practice and patience, you will become proficient in simplifying complex mathematical expressions.