Which Expression Has A Value Of 32 When $m = 8$?A. $m - 8$ B. $4m$ C. $\frac{m}{2}$ D. $m + 8$

Introduction

Algebraic expressions are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will explore the process of solving algebraic expressions, with a focus on identifying the correct expression that has a value of 32 when $m = 8$. We will examine each option carefully and provide a step-by-step solution to determine the correct answer.

Understanding Algebraic Expressions

An algebraic expression is a mathematical statement that contains variables, constants, and mathematical operations. It is a way to represent a value or a relationship between values using symbols and mathematical notation. Algebraic expressions can be simple or complex, and they can be used to solve a wide range of mathematical problems.

The Problem

We are given four algebraic expressions, and we need to determine which one has a value of 32 when $m = 8$. The expressions are:

A. $m - 8$ B. $4m$ C. $\frac{m}{2}$ D. $m + 8$

Step-by-Step Solution

To solve this problem, we will substitute $m = 8$ into each expression and evaluate the result.

Option A: $m - 8$

When we substitute $m = 8$ into the expression $m - 8$, we get:

$8 - 8 = 0$

So, the value of the expression $m - 8$ when $m = 8$ is 0.

Option B: $4m$

When we substitute $m = 8$ into the expression $4m$, we get:

$4(8) = 32$

So, the value of the expression $4m$ when $m = 8$ is 32.

Option C: $\frac{m}{2}$

When we substitute $m = 8$ into the expression $\frac{m}{2}$, we get:

$\frac{8}{2} = 4$

So, the value of the expression $\frac{m}{2}$ when $m = 8$ is 4.

Option D: $m + 8$

When we substitute $m = 8$ into the expression $m + 8$, we get:

$8 + 8 = 16$

So, the value of the expression $m + 8$ when $m = 8$ is 16.

Conclusion

Based on our step-by-step solution, we can see that only one expression has a value of 32 when $m = 8$. That expression is:

B. $4m$

Therefore, the correct answer is option B, $4m$.

Tips and Tricks

When solving algebraic expressions, it's essential to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
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.

By following this order of operations, you can ensure that you are evaluating the expressions correctly and avoiding any potential errors.

Common Mistakes

When solving algebraic expressions, it's easy to make mistakes. Here are some common mistakes to watch out for:

• Forgetting to substitute the value of the variable: Make sure to substitute the value of the variable into the expression before evaluating it.
• Not following the order of operations: Make sure to follow the order of operations (PEMDAS) to ensure that you are evaluating the expressions correctly.
• Not simplifying the expression: Make sure to simplify the expression as much as possible before evaluating it.

By avoiding these common mistakes, you can ensure that you are solving algebraic expressions correctly and accurately.

Practice Problems

To practice solving algebraic expressions, try the following problems:

1. Solve the expression $2x + 5$ when $x = 3$.
2. Solve the expression $\frac{x}{4} - 2$ when $x = 12$.
3. Solve the expression $x^2 + 4x - 5$ when $x = 2$.

By practicing these problems, you can improve your skills and become more confident in solving algebraic expressions.

Conclusion

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Q: What is an algebraic expression?

A: An algebraic expression is a mathematical statement that contains variables, constants, and mathematical operations. It is a way to represent a value or a relationship between values using symbols and mathematical notation.

Q: What are the different types of algebraic expressions?

A: There are several types of algebraic expressions, including:

• Polynomial expressions: These are expressions that consist of variables and constants, and are often used to represent quadratic equations.
• Rational expressions: These are expressions that consist of a fraction of two polynomials.
• Exponential expressions: These are expressions that involve variables raised to a power.
• Trigonometric expressions: These are expressions that involve trigonometric functions such as sine, cosine, and tangent.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you can use the following steps:

1. Combine like terms: Combine any terms that have the same variable and exponent.
2. Simplify fractions: Simplify any fractions in the expression by dividing the numerator and denominator by their greatest common divisor.
3. Simplify exponents: Simplify any exponents in the expression by combining like terms.
4. Simplify trigonometric functions: Simplify any trigonometric functions in the expression by using trigonometric identities.

Q: How do I evaluate an algebraic expression?

A: To evaluate an algebraic expression, you can use the following steps:

1. Substitute the value of the variable: Substitute the value of the variable into the expression.
2. Follow the order of operations: Follow the order of operations (PEMDAS) to ensure that you are evaluating the expression correctly.
3. Simplify the expression: Simplify the expression as much as possible before evaluating it.

Q: What is the difference between an algebraic expression and an equation?

A: An algebraic expression is a mathematical statement that contains variables, constants, and mathematical operations. An equation is a mathematical statement that states that two expressions are equal. For example, the expression $x + 2$ is an algebraic expression, while the equation $x + 2 = 5$ is an equation.

Q: How do I solve an algebraic equation?

A: To solve an algebraic equation, you can use the following steps:

1. Isolate the variable: Isolate the variable on one side of the equation.
2. Simplify the equation: Simplify the equation as much as possible.
3. Solve for the variable: Solve for the variable by using inverse operations.

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

A: Some common mistakes to avoid when solving algebraic expressions include:

• Forgetting to substitute the value of the variable: Make sure to substitute the value of the variable into the expression before evaluating it.
• Not following the order of operations: Make sure to follow the order of operations (PEMDAS) to ensure that you are evaluating the expression correctly.
• Not simplifying the expression: Make sure to simplify the expression as much as possible before evaluating it.

Q: How can I practice solving algebraic expressions?

A: You can practice solving algebraic expressions by:

• Working through practice problems: Try solving algebraic expressions on your own, using practice problems from a textbook or online resource.
• Using online resources: Use online resources such as Khan Academy, Mathway, or Wolfram Alpha to practice solving algebraic expressions.
• Seeking help from a tutor or teacher: If you are struggling to solve algebraic expressions, consider seeking help from a tutor or teacher.

Conclusion

Solving algebraic expressions is a crucial skill for students to master. By following the order of operations and avoiding common mistakes, you can ensure that you are solving expressions correctly and accurately. In this article, we explored some frequently asked questions about algebraic expressions, including what they are, how to simplify and evaluate them, and how to solve algebraic equations. By following this guide, you can improve your skills and become more confident in solving algebraic expressions.