Problem 62 - Elementary and Intermediate Algebra

Question

Multiply. $$ (2+a+b) 6 $$

Step-by-Step Solution

Answer

12 + 6a + 6b

Step 1 - Distribute the 6

Apply the distributive property to multiply each term inside the parenthesis by 6. This means multiplying 6 by 2, 6 by a, and 6 by b.

Step 2 - Perform the multiplications

Calculate the products: \[6 \times 2 = 12\]\[6 \times a = 6a\]\[6 \times b = 6b\]

Step 3 - Combine the results

Combine all the products to get the final expression: \[12 + 6a + 6b\]

Key Concepts

Multiplication

Multiplication is one of the basic operations in mathematics. It involves combining equal groups together. In the given exercise, we aim to multiply a number (6) with every term inside the parentheses (2, a, b). This method is part of the distributive property, which makes handling complex expressions simpler.

When we say 6 multiplied by 2, it's like having 6 groups of 2, which equals 12. Similarly, multiplying 6 with a variable like 'a' means we have 'a' grouped 6 times. So the result becomes 6a.

• 6 × 2 = 12
• 6 × a = 6a
• 6 × b = 6b

Combining these multiplications, we get the final result, which simplifies the expression.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are a fundamental part of algebra and help us represent problems in a generalized form.

In our exercise example, the expression (2 + a + b) is an algebraic expression within parentheses, and we are multiplying it by 6.

Here, '2' is a constant term, while 'a' and 'b' are variables. Variables stand for unknown values that can change. This process of multiplying each term in the expression by 6 modifies each part of the original expression while preserving its overall structure.

Parentheses

Parentheses are symbols used in mathematics to group terms and operations. They indicate which operations should be performed first. Parentheses help in organizing and clarifying expressions, especially when using the distributive property.

In the given exercise, the expression is (2 + a + b). Parentheses tell us to address the entire sum inside them before moving forward. When applying the distributive property, we multiply everything inside the parentheses by 6.

Here's how it works step-by-step:

• Distribute 6 to each term inside: 6 × 2, 6 × a, 6 × b
• Perform the multiplications: 12, 6a, 6b
• Combine the results: 12 + 6a + 6b

Parentheses ensure all internal operations are respected and help avoid mistakes.