y = 2(2x + 7)^3
y’ = 48x^2 + 336x + 588
The given expression is y = 2(2x + 7)^3.
To simplify this expression, we can use the rule of exponents that states:
(a + b)^n = Σ(n choose k) * a^(n-k) * b^k
Where Σ means to sum over k from 0 to n, (n choose k) is the binomial coefficient, and k is the power of b while (n-k) is the power of a.
Using this rule, we can expand (2x + 7)^3 as:
(2x + 7)^3 = Σ(3 choose k) * (2x)^(3-k) * (7)^k
= (3 choose 0) * (2x)^(3-0) * (7)^0 + (3 choose 1) * (2x)^(3-1) * (7)^1 + (3 choose 2) * (2x)^(3-2) * (7)^2 + (3 choose 3) * (2x)^(3-3) * (7)^3
= 1 * 8x^3 * 1 + 3 * 4x^2 * 7 + 3 * 2x * 49 + 1 * 343
= 8x^3 + 84x^2 + 294x + 343
Now we can substitute this expanded expression back into the original expression:
y = 2(2x + 7)^3
y = 2(8x^3 + 84x^2 + 294x + 343)
y = 16x^3 + 168x^2 + 588x + 686
Therefore, the simplified expression for y is 16x^3 + 168x^2 + 588x + 686.
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