Which Expression Is Equivalent To 36 A − 27 36a - 27 36 A − 27 ?A. 9 ( 4 A − 3 9(4a - 3 9 ( 4 A − 3 ] B. 3 ( 18 A − 9 3(18a - 9 3 ( 18 A − 9 ] C. 9 ( 4 A − 27 9(4a - 27 9 ( 4 A − 27 ] D. 3 ( 12 A − 6 3(12a - 6 3 ( 12 A − 6 ]

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given problem: Which expression is equivalent to $36a - 27$?

Understanding the Problem

The problem requires us to find an equivalent expression for $36a - 27$. This means we need to find an expression that has the same value as $36a - 27$ when evaluated. To do this, we can use the distributive property of multiplication over addition, which states that $a(b + c) = ab + ac$.

Option A: $9(4a - 3)$

Let's start by analyzing Option A: $9(4a - 3)$. Using the distributive property, we can expand this expression as follows:

$$9(4a - 3) = 9(4a) - 9(3) = 36a - 27$$

As we can see, Option A is indeed equivalent to $36a - 27$.

Option B: $3(18a - 9)$

Next, let's analyze Option B: $3(18a - 9)$. Using the distributive property, we can expand this expression as follows:

$$3(18a - 9) = 3(18a) - 3(9) = 54a - 27$$

As we can see, Option B is not equivalent to $36a - 27$.

Option C: $9(4a - 27)$

Now, let's analyze Option C: $9(4a - 27)$. Using the distributive property, we can expand this expression as follows:

$$9(4a - 27) = 9(4a) - 9(27) = 36a - 243$$

As we can see, Option C is not equivalent to $36a - 27$.

Option D: $3(12a - 6)$

Finally, let's analyze Option D: $3(12a - 6)$. Using the distributive property, we can expand this expression as follows:

$$3(12a - 6) = 3(12a) - 3(6) = 36a - 18$$

As we can see, Option D is not equivalent to $36a - 27$.

Conclusion

In conclusion, the correct answer is Option A: $9(4a - 3)$. This expression is equivalent to $36a - 27$ when evaluated. The distributive property of multiplication over addition was used to expand and simplify the expressions.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

• Use the distributive property to expand and simplify expressions.
• Look for common factors in the numerator and denominator.
• Use the commutative and associative properties of addition and multiplication to rearrange terms.
• Simplify fractions by canceling out common factors.

Practice Problems

Here are some practice problems to help you practice simplifying algebraic expressions:

• Simplify the expression: $2(3x + 4) - 5$
• Simplify the expression: $4(2x - 3) + 2$
• Simplify the expression: $3(2x + 1) - 2(3x - 2)$

Real-World Applications

Simplifying algebraic expressions has many real-world applications, including:

• Calculating the area and perimeter of shapes.
• Finding the volume of solids.
• Modeling population growth and decay.
• Solving systems of equations.

Conclusion

Introduction

In our previous article, we explored the process of simplifying algebraic expressions, with a focus on the given problem: Which expression is equivalent to $36a - 27$? In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q: What is the distributive property of multiplication over addition?

A: The distributive property of multiplication over addition states that $a(b + c) = ab + ac$. This means that when you multiply a number by a sum, you can multiply the number by each term in the sum and then add the results.

Q: How do I simplify an expression using the distributive property?

A: To simplify an expression using the distributive property, follow these steps:

1. Identify the terms in the expression that are being multiplied together.
2. Multiply each term by the number outside the parentheses.
3. Add the results together.

Q: What is the difference between the distributive property and the commutative property?

A: The distributive property states that $a(b + c) = ab + ac$, while the commutative property states that $a + b = b + a$. The commutative property allows you to rearrange the terms in an expression, while the distributive property allows you to multiply a number by a sum.

Q: How do I simplify a fraction?

A: To simplify a fraction, follow these steps:

1. Identify the numerator and denominator.
2. Look for common factors in the numerator and denominator.
3. Cancel out the common factors.

Q: What is the associative property of addition?

A: The associative property of addition states that $(a + b) + c = a + (b + c)$. This means that when you add three numbers together, you can add the first two numbers together and then add the third number, or you can add the first and third numbers together and then add the second number.

Q: How do I simplify an expression using the associative property?

A: To simplify an expression using the associative property, follow these steps:

1. Identify the terms in the expression that are being added together.
2. Rearrange the terms to group the numbers together in a way that makes it easy to add them.
3. Add the numbers together.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when you have multiple operations in an expression. The order of operations is:

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

Q: How do I apply the order of operations to simplify an expression?

A: To apply the order of operations to simplify an expression, follow these steps:

1. Identify any expressions inside parentheses and evaluate them first.
2. Identify any exponents and evaluate them next.
3. Identify any multiplication and division operations and evaluate them from left to right.
4. Finally, identify any addition and subtraction operations and evaluate them from left to right.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By using the distributive property, looking for common factors, and simplifying fractions, you can simplify even the most complex expressions. With practice and patience, you can become proficient in simplifying algebraic expressions and apply them to real-world problems.