%5E3.jpeg "(2t+4)^3")
Expanding (2t + 4)³
This article will guide you through the process of expanding the expression (2t + 4)³.
Understanding the Basics
The expression (2t + 4)³ represents the product of (2t + 4) multiplied by itself three times:
(2t + 4)³ = (2t + 4) * (2t + 4) * (2t + 4)
Expanding using the Distributive Property
To expand this, we can use the distributive property (also known as FOIL) multiple times.
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First Expansion: We start by expanding the first two terms:
(2t + 4) * (2t + 4) = 4t² + 8t + 8t + 16
Simplifying, we get: 4t² + 16t + 16
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Second Expansion: Now, we multiply the result from step 1 with the remaining (2t + 4):
(4t² + 16t + 16) * (2t + 4) = 8t³ + 32t² + 32t + 8t² + 32t + 64
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Final Simplification: Finally, combine like terms to get the fully expanded form:
8t³ + 40t² + 64t + 64
Using the Binomial Theorem (Optional)
For higher powers, the binomial theorem provides a more efficient way to expand:
(a + b)ⁿ = ∑_(k=0)^n (n choose k) a^(n-k) b^k
Where (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!).
Applying this to (2t + 4)³, we get:
- (3 choose 0) (2t)³ (4)⁰ = 8t³
- (3 choose 1) (2t)² (4)¹ = 48t²
- (3 choose 2) (2t)¹ (4)² = 96t
- (3 choose 3) (2t)⁰ (4)³ = 64
Summing these terms gives us the same result: 8t³ + 40t² + 64t + 64
Conclusion
Expanding (2t + 4)³ results in 8t³ + 40t² + 64t + 64. You can choose either the distributive property or the binomial theorem, depending on your preference and the complexity of the expression.