Explanation:
To expand and fully simplify the expression 3(4n + 5) + 6n + 4, we can follow the order of operations, which is also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).
First, let's simplify the expression inside the parentheses: 4n + 5. To do this, we distribute the 3 to both terms inside the parentheses:
3 * 4n + 3 * 5
This gives us:
12n + 15
Now, let's substitute this simplified expression back into the original expression:
3(12n + 15) + 6n + 4
Next, we distribute the 3 to both terms inside the parentheses:
3 * 12n + 3 * 15 + 6n + 4
This gives us:
36n + 45 + 6n + 4
Now, let's combine like terms. We add the coefficients of the n terms and the constants:
(36n + 6n) + (45 + 4)
This gives us:
42n + 49
Therefore, the expanded and fully simplified form of 3(4n + 5) + 6n + 4 is 42n + 49.
Side Note(Explanation): When simplifying expressions with parentheses, it is important to follow the order of operations, also known as PEMDAS. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
The purpose of the order of operations is to provide a systematic way to simplify expressions and ensure that everyone arrives at the same answer. Let's break down each step:
1. Parentheses: Start by simplifying any expressions inside parentheses. This involves performing any operations within the parentheses first. For example, if you have an expression like 3(4 + 2), you would first add 4 and 2 to get 6, and then multiply 3 by 6 to get the final answer of 18.
2. Exponents: If there are any exponents (numbers raised to a power), simplify them next. For example, if you have an expression like 2^3, you would calculate 2 raised to the power of 3, which equals 8.
3. Multiplication and Division: After simplifying any parentheses and exponents, perform any multiplication and division operations from left to right. For example, if you have an expression like 4 * 2 / 2, you would first multiply 4 by 2 to get 8, and then divide 8 by 2 to get the final answer of 4.
4. Addition and Subtraction: Finally, perform any addition and subtraction operations from left to right. For example, if you have an expression like 5 + 3 - 2, you would first add 5 and 3 to get 8, and then subtract 2 to get the final answer of 6.
By following the order of operations, you ensure that each operation is performed in the correct sequence, leading to the correct answer. It is important to note that parentheses can be nested within each other, and in such cases, you would start by simplifying the innermost parentheses first and work your way outwards.
Now, let's move on to some practice problems to solidify our understanding.
Practice problem 1:
Expand and fully simplify the expression 2(3x + 4) + 5x + 2.
Practice problem 2:
Expand and fully simplify the expression 4(2y - 3) + 7y - 5.
Practice problem 3:
Expand and fully simplify the expression 5(2z + 1) + 3z - 2.
Try solving these problems on your own, and let me know if you need any further assistance!