What is the term independent of x in the expansion of (3x - 4)^7?

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Question: What is the term independent of x in the expansion of (3x - 4)^7?

Options:

Correct Answer: 84

Solution:

The term independent of x occurs when k = 7, C(7, 3) * (3)^3 * (-4)^4 = 35 * 27 * 256 = 84.

What is the term independent of x in the expansion of (3x - 4)^7?

What is the term independent of x in the expansion of (3x - 4)^7?

Step 1: Identify the expression to expand, which is (3x - 4)^7.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = Σ (C(n, k) * a^(n-k) * b^k) for k = 0 to n.
Step 3: In our case, a = 3x, b = -4, and n = 7.
Step 4: We need to find the term that does not contain x, which means we want the term where the power of x is 0.
Step 5: The term in the expansion is given by C(7, k) * (3x)^(7-k) * (-4)^k.
Step 6: Set the exponent of x to 0: (3x)^(7-k) = 0 when 7-k = 0, which means k = 7.
Step 7: Substitute k = 7 into the term: C(7, 7) * (3x)^(7-7) * (-4)^7.
Step 8: Calculate C(7, 7), which is 1, and (3x)^0, which is 1.
Step 9: Now calculate (-4)^7, which is -16384.
Step 10: The term independent of x is -16384.

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