3.jpeg "(4/5a-2)3")
Simplifying the Expression (4/5a - 2)^3
This expression represents a cube of a binomial, which is a sum or difference of two terms. To simplify it, we need to expand it using the binomial theorem.
The Binomial Theorem
The binomial theorem provides a formula for expanding expressions of the form (x + y)^n:
(x + y)^n = x^n + (n choose 1)x^(n-1)y + (n choose 2)x^(n-2)y^2 + ... + (n choose n-1)xy^(n-1) + y^n
Where (n choose k) represents the binomial coefficient, which is calculated as:
(n choose k) = n! / (k! * (n-k)!)
Applying the Theorem to (4/5a - 2)^3
In our case, x = 4/5a, y = -2, and n = 3. Let's apply the binomial theorem step by step:
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First term: (4/5a)^3 = 64/125a^3
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Second term: (3 choose 1)(4/5a)^2(-2) = -96/25a^2
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Third term: (3 choose 2)(4/5a)(-2)^2 = 48/5a
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Fourth term: (3 choose 3)(-2)^3 = -8
Final Expansion
Putting all the terms together, we get the simplified expansion:
(4/5a - 2)^3 = 64/125a^3 - 96/25a^2 + 48/5a - 8
Conclusion
By applying the binomial theorem, we were able to expand and simplify the expression (4/5a - 2)^3. The resulting expression is a polynomial with four terms, each with a specific coefficient and power of 'a'.