What is the value of the term containing x^4 in the expansion of (3x + 4)^6? (19

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Question: What is the value of the term containing x^4 in the expansion of (3x + 4)^6? (1944)

Options:

Correct Answer: 1296

Exam Year: 1944

Solution:

The term containing x^4 is given by C(6, 4)(3x)^4(4)^2 = 15 * 81 * 16 = 19440.

What is the value of the term containing x^4 in the expansion of (3x + 4)^6? (1944)

What is the value of the term containing x^4 in the expansion of (3x + 4)^6? (1944)

Step 1: Identify the expression to expand, which is (3x + 4)^6.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = Σ C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient.
Step 3: In our case, a = 3x, b = 4, and n = 6.
Step 4: We want the term that contains x^4. This occurs when (3x) is raised to the power of 4.
Step 5: Determine k, the power of b (which is 4). Since n = 6, we have k = 6 - 4 = 2.
Step 6: Calculate the binomial coefficient C(6, 4), which is equal to C(6, 2) = 15.
Step 7: Calculate (3x)^4, which is (3^4)(x^4) = 81x^4.
Step 8: Calculate 4^2, which is 16.
Step 9: Combine these values to find the term: C(6, 4) * (3x)^4 * (4^2) = 15 * 81 * 16.
Step 10: Calculate 15 * 81 = 1215, then multiply by 16 to get 19440.

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