Simplify the Expressions and State Whether They Are True or False

Oct 13, 2025 by miodeya.com 26 views

Simplify the Expressions and State Whether They Are True or False

In mathematics, simplifying expressions is a crucial skill that helps us evaluate the truth of various statements. In this article, we will focus on simplifying two expressions involving division and fractions. We will use the correct order of operations to simplify the expressions and then determine whether they are true or false.

The first expression is $4 \div \frac{2}{3} = 6$ . To simplify this expression, we need to follow the order of operations, which states that division comes before multiplication and addition. In this case, we have a division operation involving a fraction.

To simplify the expression, we can use the following steps:

Invert and Multiply: When dividing by a fraction, we can invert the fraction and multiply instead. In this case, we can invert $\frac{2}{3}$ to get $\frac{3}{2}$ .
Multiply: Now, we can multiply 4 by $\frac{3}{2}$ to get $\frac{12}{2}$ .
Simplify: Finally, we can simplify $\frac{12}{2}$ to get 6.

Therefore, the simplified expression is $4 \div \frac{2}{3} = 6$ . However, we need to determine whether this expression is true or false.

Is Expression A True or False?

To determine whether expression A is true or false, we need to evaluate the original statement. The original statement is $4 \div \frac{2}{3} = 6$ . We have already simplified the expression to get 6, but we need to check whether this is equal to the original value.

Let's evaluate the original expression:

$4 \div \frac{2}{3} = 4 \times \frac{3}{2} = 6$

As we can see, the original expression is indeed equal to 6. Therefore, expression A is TRUE.

The second expression is $4 \div \frac{1}{3} = 12$ . To simplify this expression, we can use the same steps as before:

Invert and Multiply: When dividing by a fraction, we can invert the fraction and multiply instead. In this case, we can invert $\frac{1}{3}$ to get $\frac{3}{1}$ .
Multiply: Now, we can multiply 4 by $\frac{3}{1}$ to get $\frac{12}{1}$ .
Simplify: Finally, we can simplify $\frac{12}{1}$ to get 12.

Therefore, the simplified expression is $4 \div \frac{1}{3} = 12$ . However, we need to determine whether this expression is true or false.

Is Expression B True or False?

To determine whether expression B is true or false, we need to evaluate the original statement. The original statement is $4 \div \frac{1}{3} = 12$ . We have already simplified the expression to get 12, but we need to check whether this is equal to the original value.

Let's evaluate the original expression:

$4 \div \frac{1}{3} = 4 \times \frac{3}{1} = 12$

As we can see, the original expression is indeed equal to 12. Therefore, expression B is TRUE.

In this article, we simplified two expressions involving division and fractions. We used the correct order of operations to simplify the expressions and then determined whether they were true or false. We found that both expressions were true, as the simplified expressions were equal to the original values.

Key Takeaways

When dividing by a fraction, we can invert the fraction and multiply instead.
The order of operations states that division comes before multiplication and addition.
Simplifying expressions is a crucial skill that helps us evaluate the truth of various statements.

Final Thoughts

Simplifying expressions is an essential skill in mathematics that helps us evaluate the truth of various statements. By following the correct order of operations and using the correct techniques, we can simplify complex expressions and determine whether they are true or false.
Simplify the Expressions and State Whether They Are True or False: Q&A

In our previous article, we simplified two expressions involving division and fractions. We used the correct order of operations to simplify the expressions and then determined whether they were true or false. In this article, we will provide a Q&A section to help you better understand the concepts and techniques used in simplifying expressions.

Q: What is the correct order of operations when simplifying expressions? A: The correct order of operations is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate 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 an expression involving division and a fraction? A: To simplify an expression involving division and a fraction, you can use the following steps:

Invert and Multiply: When dividing by a fraction, you can invert the fraction and multiply instead.
Multiply: Multiply the numerator of the fraction by the dividend.
Simplify: Simplify the resulting expression.

Q: What is the difference between a fraction and a decimal? A: A fraction is a way of expressing a part of a whole, while a decimal is a way of expressing a fraction as a numerical value. For example, the fraction $\frac{1}{2}$ is equal to the decimal 0.5.

Q: How do I determine whether an expression is true or false? A: To determine whether an expression is true or false, you need to evaluate the expression and compare the result to the original value. If the result is equal to the original value, then the expression is true. If the result is not equal to the original value, then the expression is false.

Q: What are some common mistakes to avoid when simplifying expressions? A: Some common mistakes to avoid when simplifying expressions include:

Not following the correct order of operations
Not inverting and multiplying when dividing by a fraction
Not simplifying the resulting expression
Not evaluating the expression correctly

Q: How can I practice simplifying expressions? A: You can practice simplifying expressions by working through examples and exercises. You can also use online resources and tools to help you practice and improve your skills.

Q: What are some real-world applications of simplifying expressions? A: Simplifying expressions has many real-world applications, including:

Calculating costs and prices
Determining the area and perimeter of shapes
Evaluating the performance of systems and machines
Solving problems in science, technology, engineering, and mathematics (STEM) fields

In this article, we provided a Q&A section to help you better understand the concepts and techniques used in simplifying expressions. We covered topics such as the correct order of operations, simplifying expressions involving division and fractions, and determining whether an expression is true or false. We also discussed common mistakes to avoid and provided tips for practicing and improving your skills.

Key Takeaways

The correct order of operations is: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
To simplify an expression involving division and a fraction, you can use the steps: Invert and Multiply, Multiply, and Simplify.
To determine whether an expression is true or false, you need to evaluate the expression and compare the result to the original value.
Common mistakes to avoid when simplifying expressions include not following the correct order of operations, not inverting and multiplying when dividing by a fraction, and not simplifying the resulting expression.

Final Thoughts

Simplifying expressions is an essential skill in mathematics that has many real-world applications. By following the correct order of operations and using the correct techniques, you can simplify complex expressions and determine whether they are true or false. With practice and patience, you can become proficient in simplifying expressions and apply your skills to a wide range of problems and situations.