Question

rewrite the following expression in expanded form and simplify completely. $(2x + 4y)^5$

Explanation:

Step1: Apply Binomial Theorem

The binomial theorem states:
$$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}$$
For $(2x+4y)^5$, $a=2x$, $b=4y$, $n=5$.

Step2: Expand each term

Calculate each term of the sum:

• $k=0$: $\binom{5}{0}(2x)^5(4y)^0 = 1 \cdot 32x^5 \cdot 1 = 32x^5$
• $k=1$: $\binom{5}{1}(2x)^4(4y)^1 = 5 \cdot 16x^4 \cdot 4y = 320x^4y$
• $k=2$: $\binom{5}{2}(2x)^3(4y)^2 = 10 \cdot 8x^3 \cdot 16y^2 = 1280x^3y^2$
• $k=3$: $\binom{5}{3}(2x)^2(4y)^3 = 10 \cdot 4x^2 \cdot 64y^3 = 2560x^2y^3$
• $k=4$: $\binom{5}{4}(2x)^1(4y)^4 = 5 \cdot 2x \cdot 256y^4 = 2560xy^4$
• $k=5$: $\binom{5}{5}(2x)^0(4y)^5 = 1 \cdot 1 \cdot 1024y^5 = 1024y^5$

Step3: Combine all terms

Sum the calculated terms together.

Answer:

$32x^5 + 320x^4y + 1280x^3y^2 + 2560x^2y^3 + 2560xy^4 + 1024y^5$