Verified Answer :

  • Use the distributive property \(a(b+c) = ab + ac\) to distribute the 8 across the terms inside the parentheses.
  • Multiply 8 by \(3x\) to get \(24x\).
  • Multiply 8 by 4 to get \(32\).
  • Combine the results to get \(24x + 32\), which matches option \(\boxed{a}\).

Explanation

  1. Understanding the Distributive Property

To simplify the expression \(8(3x + 4)\), we use the distributive property of multiplication. The distributive property states that for any numbers \(a\), \(b\), and \(c\), the product \(a(b + c)\) is equal to \(ab + ac\). In our case, \(a = 8\), \(b = 3x\), and \(c = 4\).

  1. Applying the Property and Simplifying

By applying the distributive property, we multiply the term outside the parentheses, which is 8, by each term inside the parentheses separately:
\[8(3x + 4) = 8 \cdot (3x) + 8 \cdot 4\]
Now, we calculate each product:

  • First term: \(8 \times 3x = 24x\)
  • Second term: \(8 \times 4 = 32\)

Thus, the simplified expression is \(24x + 32\).

  1. Selecting the Correct Option

We now compare our calculated result, \(24x + 32\), with the given options:

A\(24x + 32\)
B\(96x\)
C\(11x + 12\)
D\(24x + 4\)

Our result matches option a. Therefore, the simplified expression is \(24x + 32\), which corresponds to option a. The final answer is \(\boxed{a}\).

Examples

The distributive property is a fundamental tool for simplifying complex expressions and solving algebraic equations. For example, if you are calculating the total cost of a shopping trip where you buy 8 sets of an item consisting of a \$3x toy and a \$4 candy bar, you would calculate the total as \(8(3x + 4) = 24x + 32\). This skill allows you to quickly evaluate expressions in fields ranging from retail and finance to science and engineering, making mathematical modeling and problem-solving more efficient.

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