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Simplify the Algebraic Expression 7(n + 2(n + 3k) + 7k)
Mathematics
Grade 7 (Junior High School)
Question Content
Perform the indicated operations and simplify: 7(n + 2(n + 3k) + 7k)
Correct Answer
21n + 147k
Detailed Solution Steps
1
Step 1: First, simplify the innermost parentheses by applying the distributive property to 2(n + 3k). Multiply 2 by each term inside the parentheses: 2*n + 2*3k = 2n + 6k.
2
Step 2: Substitute the result back into the original expression, so it becomes 7(n + 2n + 6k + 7k).
3
Step 3: Combine like terms inside the outer parentheses. Combine the n terms: n + 2n = 3n. Combine the k terms: 6k + 7k = 13k. Now the expression is 7(3n + 13k).
4
Step 4: Apply the distributive property again, multiplying 7 by each term inside the parentheses: 7*3n + 7*13k = 21n + 91k.
Knowledge Points Involved
1
Distributive Property of Multiplication over Addition
This property states that a(b + c) = ab + ac, where a, b, and c are real numbers or algebraic terms. It is used to expand expressions by multiplying a single term across the sum or difference inside parentheses, which is essential for simplifying algebraic expressions with nested operations.
2
Combining Like Terms
Like terms are algebraic terms that have the same variables raised to the same powers. To combine them, add or subtract their coefficients while keeping the variable part unchanged. For example, 2n + n = 3n and 6k + 7k = 13k. This step simplifies expressions by reducing the number of terms.
3
Order of Operations (PEMDAS/BODMAS)
This rule dictates the sequence of solving mathematical operations: Parentheses/Brackets first, then Exponents/Orders, followed by Multiplication and Division, and finally Addition and Subtraction. For nested parentheses, solve the innermost set first, which is critical for simplifying complex algebraic expressions correctly.
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