What is the term containing x^5 in the expansion of (2x + 3)^6?

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Question: What is the term containing x^5 in the expansion of (2x + 3)^6?

Options:

Correct Answer: 720

Solution:

The term containing x^5 occurs when k = 5. Using the binomial theorem, the term is C(6,5) * (2x)^5 * 3^1 = 6 * 32 * 3 = 576.

What is the term containing x^5 in the expansion of (2x + 3)^6?

What is the term containing x^5 in the expansion of (2x + 3)^6?

Step 1: Identify the expression we are expanding, which is (2x + 3)^6.
Step 2: Understand that we want the term that contains x^5.
Step 3: Use the binomial theorem, which states that the expansion of (a + b)^n is given by the sum of terms of the form C(n, k) * a^(n-k) * b^k.
Step 4: In our case, a = 2x, b = 3, and n = 6.
Step 5: To find the term with x^5, we need to set k = 5 because (2x)^(n-k) will give us x^5 when n-k = 5.
Step 6: Calculate n-k: 6 - 5 = 1, so we will have 3^1 in our term.
Step 7: Calculate the coefficient using C(6, 5), which is the number of ways to choose 5 from 6. This equals 6.
Step 8: Now calculate the term: C(6, 5) * (2x)^5 * 3^1.
Step 9: Calculate (2x)^5: (2^5) * (x^5) = 32 * x^5.
Step 10: Now substitute back into the term: 6 * (32 * x^5) * 3.
Step 11: Calculate 6 * 32 * 3 = 576.
Step 12: The term containing x^5 in the expansion is 576.

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