Evaluate: \[$(4 + 6 \cdot 3) + 3\$\]A. 33 B. 66 C. 16 D. 25
Understanding the Order of Operations
When evaluating mathematical expressions, it's essential to follow the order of operations, which is a set of rules that dictate the order in which operations should be performed. The order of operations is often remembered using the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
Evaluating the Expression
The given expression is {(4 + 6 \cdot 3) + 3$}$. To evaluate this expression, we need to follow the order of operations.
Step 1: Evaluate the Expression Inside the Parentheses
The expression inside the parentheses is ${4 + 6 \cdot 3\$}. According to the order of operations, we need to evaluate the multiplication first.
Step 2: Multiply 6 and 3
The product of 6 and 3 is 18.
Step 3: Add 4 and 18
The sum of 4 and 18 is 22.
Step 4: Add 3 to the Result
The final result is 22 + 3 = 25.
Conclusion
Based on the order of operations, the correct answer is D. 25.
Why is the Order of Operations Important?
The order of operations is essential in mathematics because it helps to avoid confusion and ensure that mathematical expressions are evaluated consistently. Without the order of operations, mathematical expressions could be evaluated differently depending on the individual performing the calculation.
Real-World Applications of the Order of Operations
The order of operations has numerous real-world applications, including:
- Computer Programming: The order of operations is crucial in computer programming, where mathematical expressions are used to perform calculations and make decisions.
- Science and Engineering: The order of operations is essential in scientific and engineering applications, where mathematical expressions are used to model and analyze complex systems.
- Finance: The order of operations is important in finance, where mathematical expressions are used to calculate interest rates, investments, and other financial metrics.
Common Mistakes to Avoid
When evaluating mathematical expressions, it's essential to avoid common mistakes, including:
- Not following the order of operations: Failing to follow the order of operations can lead to incorrect results.
- Not using parentheses correctly: Using parentheses incorrectly can lead to confusion and incorrect results.
- Not evaluating expressions from left to right: Failing to evaluate expressions from left to right can lead to incorrect results.
Tips for Evaluating Mathematical Expressions
To evaluate mathematical expressions correctly, follow these tips:
- Read the expression carefully: Read the expression carefully to ensure that you understand what is being asked.
- Follow the order of operations: Follow the order of operations to ensure that mathematical expressions are evaluated consistently.
- Use parentheses correctly: Use parentheses correctly to avoid confusion and ensure that mathematical expressions are evaluated correctly.
- Evaluate expressions from left to right: Evaluate expressions from left to right to ensure that mathematical expressions are evaluated consistently.
Conclusion
In conclusion, the order of operations is essential in mathematics, and it's crucial to follow the order of operations to ensure that mathematical expressions are evaluated consistently. By following the order of operations and using parentheses correctly, you can avoid common mistakes and ensure that mathematical expressions are evaluated correctly.
Q: What is the order of operations?
A: The order of operations is a set of rules that dictate the order in which operations should be performed when evaluating mathematical expressions. The order of operations is often remembered using the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
Q: Why is the order of operations important?
A: The order of operations is essential in mathematics because it helps to avoid confusion and ensure that mathematical expressions are evaluated consistently. Without the order of operations, mathematical expressions could be evaluated differently depending on the individual performing the calculation.
Q: What is the difference between multiplication and division?
A: Multiplication and division are both operations that involve numbers, but they are performed in different ways. Multiplication involves multiplying two or more numbers together, while division involves dividing one number by another.
Q: How do I evaluate an expression with multiple operations?
A: To evaluate an expression with multiple operations, follow the order of operations. First, evaluate any expressions inside parentheses. Next, evaluate any exponents. Finally, evaluate any multiplication and division operations from left to right, and then evaluate any addition and subtraction operations from left to right.
Q: What is the difference between an exponent and a power?
A: An exponent and a power are both ways of representing repeated multiplication, but they are used in different contexts. An exponent is a small number that is raised to a power, while a power is the result of raising a number to a certain exponent.
Q: How do I evaluate an expression with a negative exponent?
A: To evaluate an expression with a negative exponent, follow the rule that a negative exponent means taking the reciprocal of the base number. For example, if you have the expression 2^(-3), you would evaluate it as 1/2^3.
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, while a constant is a value that does not change.
Q: How do I evaluate an expression with a variable?
A: To evaluate an expression with a variable, you need to know the value of the variable. If you know the value of the variable, you can substitute it into the expression and evaluate it as you would any other expression.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.
Q: How do I evaluate an expression with a linear equation?
A: To evaluate an expression with a linear equation, follow the order of operations and solve for the variable.
Q: What is the difference between a rational expression and an irrational expression?
A: A rational expression is an expression that can be written as the ratio of two integers, while an irrational expression is an expression that cannot be written as the ratio of two integers.
Q: How do I evaluate an expression with a rational expression?
A: To evaluate an expression with a rational expression, follow the order of operations and simplify the expression.
Q: What is the difference between a polynomial expression and a non-polynomial expression?
A: A polynomial expression is an expression that can be written as the sum of terms, each of which is a product of a variable and a constant, while a non-polynomial expression is an expression that cannot be written in this form.
Q: How do I evaluate an expression with a polynomial expression?
A: To evaluate an expression with a polynomial expression, follow the order of operations and simplify the expression.
Q: What is the difference between a trigonometric expression and a non-trigonometric expression?
A: A trigonometric expression is an expression that involves trigonometric functions, such as sine and cosine, while a non-trigonometric expression is an expression that does not involve trigonometric functions.
Q: How do I evaluate an expression with a trigonometric expression?
A: To evaluate an expression with a trigonometric expression, follow the order of operations and simplify the expression using trigonometric identities.
Q: What is the difference between a logarithmic expression and a non-logarithmic expression?
A: A logarithmic expression is an expression that involves logarithmic functions, such as the logarithm of a number, while a non-logarithmic expression is an expression that does not involve logarithmic functions.
Q: How do I evaluate an expression with a logarithmic expression?
A: To evaluate an expression with a logarithmic expression, follow the order of operations and simplify the expression using logarithmic identities.
Q: What is the difference between a exponential expression and a non-exponential expression?
A: An exponential expression is an expression that involves exponential functions, such as the exponential of a number, while a non-exponential expression is an expression that does not involve exponential functions.
Q: How do I evaluate an expression with an exponential expression?
A: To evaluate an expression with an exponential expression, follow the order of operations and simplify the expression using exponential identities.
Q: What is the difference between a absolute value expression and a non-absolute value expression?
A: An absolute value expression is an expression that involves absolute value functions, such as the absolute value of a number, while a non-absolute value expression is an expression that does not involve absolute value functions.
Q: How do I evaluate an expression with an absolute value expression?
A: To evaluate an expression with an absolute value expression, follow the order of operations and simplify the expression using absolute value identities.
Q: What is the difference between a complex expression and a non-complex expression?
A: A complex expression is an expression that involves complex numbers, such as numbers with both real and imaginary parts, while a non-complex expression is an expression that does not involve complex numbers.
Q: How do I evaluate an expression with a complex expression?
A: To evaluate an expression with a complex expression, follow the order of operations and simplify the expression using complex number identities.
Q: What is the difference between a matrix expression and a non-matrix expression?
A: A matrix expression is an expression that involves matrices, such as arrays of numbers, while a non-matrix expression is an expression that does not involve matrices.
Q: How do I evaluate an expression with a matrix expression?
A: To evaluate an expression with a matrix expression, follow the order of operations and simplify the expression using matrix identities.
Q: What is the difference between a vector expression and a non-vector expression?
A: A vector expression is an expression that involves vectors, such as arrays of numbers, while a non-vector expression is an expression that does not involve vectors.
Q: How do I evaluate an expression with a vector expression?
A: To evaluate an expression with a vector expression, follow the order of operations and simplify the expression using vector identities.
Q: What is the difference between a differential equation and a non-differential equation?
A: A differential equation is an equation that involves derivatives, such as rates of change, while a non-differential equation is an equation that does not involve derivatives.
Q: How do I evaluate an expression with a differential equation?
A: To evaluate an expression with a differential equation, follow the order of operations and simplify the expression using differential equation identities.
Q: What is the difference between a partial differential equation and a non-partial differential equation?
A: A partial differential equation is an equation that involves partial derivatives, such as rates of change with respect to multiple variables, while a non-partial differential equation is an equation that does not involve partial derivatives.
Q: How do I evaluate an expression with a partial differential equation?
A: To evaluate an expression with a partial differential equation, follow the order of operations and simplify the expression using partial differential equation identities.
Q: What is the difference between a stochastic differential equation and a non-stochastic differential equation?
A: A stochastic differential equation is an equation that involves random variables, such as chance events, while a non-stochastic differential equation is an equation that does not involve random variables.
Q: How do I evaluate an expression with a stochastic differential equation?
A: To evaluate an expression with a stochastic differential equation, follow the order of operations and simplify the expression using stochastic differential equation identities.
Q: What is the difference between a deterministic equation and a non-deterministic equation?
A: A deterministic equation is an equation that has a unique solution, while a non-deterministic equation is an equation that has multiple solutions.
Q: How do I evaluate an expression with a deterministic equation?
A: To evaluate an expression with a deterministic equation, follow the order of operations and simplify the expression using deterministic equation identities.
Q: What is the difference between a nonlinear equation and a non-nonlinear equation?
A: A nonlinear equation is an equation that does not have a linear relationship between the variables, while a non-nonlinear equation is an equation that has a linear relationship between the variables.
Q: How do I evaluate an expression with a nonlinear equation?
A: To evaluate an expression with a nonlinear equation, follow the order of operations and simplify the expression using nonlinear equation identities.
Q: What is the difference between a homogeneous equation and a non-homogeneous equation?
A: A homogeneous equation is an equation that has the same degree on both sides, while a non-homogeneous equation is an equation that does not have the same degree on both sides.
Q: How do I evaluate an expression with a homogeneous equation?
A: To evaluate an expression with a homogeneous equation, follow the order of operations and simplify the expression using homogeneous equation identities.
Q: What is the difference between a separable equation and a non-separable equation?
A: A separable equation