Expand the expression \((2x - 3) ^6\).

Question

Using Binomial Theorem, evaluate \((102)^5\).

Answer 1

Given:

(2x − 3)⁶

Step 1: Use Binomial Theorem

(a − b)⁶ = a⁶ − 6a⁵b + 15a⁴b² − 20a³b³ + 15a²b⁴ − 6ab⁵ + b⁶

Let
a = 2x, b = 3

Step 2: Substitute and simplify each term

a⁶ = (2x)⁶ = 64x⁶

6a⁵b = 6·(2x)⁵·3 = 576x⁵

15a⁴b² = 15·(2x)⁴·9 = 2160x⁴

20a³b³ = 20·(2x)³·27 = 4320x³

15a²b⁴ = 15·(2x)²·81 = 4860x²

6ab⁵ = 6·(2x)·243 = 2916x

b⁶ = 729

Step 3: Write the expansion

(2x − 3)⁶ =

64x⁶ − 576x⁵ + 2160x⁴ − 4320x³ + 4860x² − 2916x + 729

Final Answer:

(2x − 3)⁶ = 64x⁶ − 576x⁵ + 2160x⁴ − 4320x³ + 4860x² − 2916x + 729

Solution and Explanation