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The expanded form of \( (3x + y)^3 \) is \( 27x^{3} + 27x^{2}y + 9xy^{2} + y^{3} \).
Solution
Calculate or simplify the expression \( (3x+y)^3 \).
Calculate or simplify the expression \( (3x+y)^3 \).
Simplify the expression by following steps:
- step0: Evaluate the power:
\(\left(3x+y\right)^{3}\)
- step1: Expand the expression:
\(\left(3x\right)^{3}+3\left(3x\right)^{2}y+3\times 3xy^{2}+y^{3}\)
- step2: Calculate:
\(27x^{3}+27x^{2}y+9xy^{2}+y^{3}\)
The expanded form of \((3x+y)^3\) is \(27x^{3}+27x^{2}y+9xy^{2}+y^{3}\).
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Simplify this solution The Deep Dive
To expand \( (3x + y)^3 \), we can use the binomial theorem, which tells us that \( (a + b)^n \) can be expanded into a sum involving binomial coefficients. In this case, \( a = 3x \), \( b = y \), and \( n = 3 \). The expansion gives us \( (3x)^3 + 3(3x)^2y + 3(3x)y^2 + y^3 \), which simplifies to \( 27x^3 + 27x^2y + 9xy^2 + y^3 \). If you want to understand the concepts of binomial expansion further, it's like a fun recipe where you mix different powers of your chosen ingredients! In this case, you're mixing \( (3x) \) and \( y \) in various ways, ultimately creating a delightful polynomial treat.
