Question

use pascals triangle to expand $(2 + 3y)^{4}$. $(2 + 3y)^{4}=\square$

Explanation:

Step1: Find the coefficients from Pascal's Triangle

For the power of \(4\), the coefficients from Pascal's Triangle are \(1,4,6,4,1\).

Step2: Apply the binomial expansion formula \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\)

Here \(a = 2\), \(b=3y\), \(n = 4\)

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Step3: Calculate each term

• Term 1: \(1\times(2)^{4}(3y)^{0}=1\times16\times1 = 16\)
• Term 2: \(4\times(2)^{3}(3y)^{1}=4\times8\times3y=96y\)
• Term 3: \(6\times(2)^{2}(3y)^{2}=6\times4\times9y^{2}=216y^{2}\)
• Term 4: \(4\times(2)^{1}(3y)^{3}=4\times2\times27y^{3}=216y^{3}\)
• Term 5: \(1\times(2)^{0}(3y)^{4}=1\times1\times81y^{4}=81y^{4}\)

Step4: Combine the terms

\((2 + 3y)^{4}=16+96y + 216y^{2}+216y^{3}+81y^{4}\)

Answer:

\(16 + 96y+216y^{2}+216y^{3}+81y^{4}\)