What is the term independent of x in the expansion of (2x - 3)^8?
Practice Questions
Q1
What is the term independent of x in the expansion of (2x - 3)^8?
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256
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512
Questions & Step-by-Step Solutions
What is the term independent of x in the expansion of (2x - 3)^8?
Step 1: Identify the expression we are expanding, which is (2x - 3)^8.
Step 2: Understand that we want to find the term that does not contain 'x'. This is called the term independent of x.
Step 3: 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 4: In our case, a = 2x, b = -3, and n = 8.
Step 5: We need to find the value of k where the power of x becomes zero. This happens when the power of (2x) is 4, because (2x)^4 = 16x^4, and we want x^0.
Step 6: Set up the equation: n - k = 4, which means k = 8 - 4 = 4.
Step 7: Calculate the coefficient using C(8, 4), which is the number of ways to choose 4 from 8. C(8, 4) = 70.
Step 8: Calculate (2x)^4, which is (2^4)(x^4) = 16x^4.
Step 9: Calculate (-3)^4, which is 81.
Step 10: Combine these values to find the term: C(8, 4) * (2x)^4 * (-3)^4 = 70 * 16 * 81.
