Question

expand the expression:

\\((4x + 2)^6 =\\)

Explanation:

Step1: Apply the Binomial Theorem

$$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$

Step2: Substitute the given values

$$(4x + 2)^6 = \sum_{k=0}^{6} \binom{6}{k} (4x)^{6-k} (2)^k$$

Step3: Calculate each term individually

$$\binom{6}{0}(4x)^6(2)^0 = 1 \cdot 4096x^6 \cdot 1 = 4096x^6$$

$$\binom{6}{1}(4x)^5(2)^1 = 6 \cdot 1024x^5 \cdot 2 = 12288x^5$$

$$\binom{6}{2}(4x)^4(2)^2 = 15 \cdot 256x^4 \cdot 4 = 15360x^4$$

$$\binom{6}{3}(4x)^3(2)^3 = 20 \cdot 64x^3 \cdot 8 = 10240x^3$$

$$\binom{6}{4}(4x)^2(2)^4 = 15 \cdot 16x^2 \cdot 16 = 3840x^2$$

$$\binom{6}{5}(4x)^1(2)^5 = 6 \cdot 4x \cdot 32 = 768x$$

$$\binom{6}{6}(4x)^0(2)^6 = 1 \cdot 1 \cdot 64 = 64$$

Step4: Combine all the calculated terms

$$4096x^6 + 12288x^5 + 15360x^4 + 10240x^3 + 3840x^2 + 768x + 64$$

Answer:

\(4096x^6 + 12288x^5 + 15360x^4 + 10240x^3 + 3840x^2 + 768x + 64\)