Find the value of (3 + 2)^3 using the binomial theorem.

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Question: Find the value of (3 + 2)^3 using the binomial theorem.

Options:

Correct Answer: 27

Solution:

Using the binomial theorem, (3 + 2)^3 = C(3,0) * 3^3 * 2^0 + C(3,1) * 3^2 * 2^1 + C(3,2) * 3^1 * 2^2 + C(3,3) * 3^0 * 2^3 = 27 + 54 + 36 + 8 = 125.

Find the value of (3 + 2)^3 using the binomial theorem.

Find the value of (3 + 2)^3 using the binomial theorem.

Step 1: Identify the expression we want to calculate, which is (3 + 2)^3.
Step 2: Recognize that we can 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 = 3, b = 2, and n = 3.
Step 4: Write out the binomial expansion for (3 + 2)^3 using the formula: C(3,0) * 3^3 * 2^0 + C(3,1) * 3^2 * 2^1 + C(3,2) * 3^1 * 2^2 + C(3,3) * 3^0 * 2^3.
Step 5: Calculate each term separately: C(3,0) = 1, C(3,1) = 3, C(3,2) = 3, C(3,3) = 1.
Step 6: Calculate the first term: C(3,0) * 3^3 * 2^0 = 1 * 27 * 1 = 27.
Step 7: Calculate the second term: C(3,1) * 3^2 * 2^1 = 3 * 9 * 2 = 54.
Step 8: Calculate the third term: C(3,2) * 3^1 * 2^2 = 3 * 3 * 4 = 36.
Step 9: Calculate the fourth term: C(3,3) * 3^0 * 2^3 = 1 * 1 * 8 = 8.
Step 10: Add all the terms together: 27 + 54 + 36 + 8 = 125.
Step 11: Conclude that the value of (3 + 2)^3 is 125.

Binomial Theorem – The binomial theorem provides a formula for expanding expressions of the form (a + b)^n, where n is a non-negative integer.
Combinatorial Coefficients – The coefficients C(n, k) represent the number of ways to choose k elements from a set of n elements, which are used in the expansion.
Exponents and Powers – Understanding how to calculate powers of numbers and apply them in the context of the binomial expansion.