How do you use the binomial theorem to expand and simplify the expression $(a+6)^4$ ?

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Answer 1 · 2017-08-27T15:51:40

$a^4 + 24a^3 + 216a^2 + 864a + 1296$

$(a+b)^n = ((n),(0)) a^n b^0 + ((n),(1)) a^(n-1) b^1 + ((n),(2)) a^(n-2) b^2 + ... + ((n),(n)) a^0 b^n$

$((n),(r))$ is a combination read as " $n$ choose $r$ ".

$((n),(r)) = color(white)I_nC_r = (n!) / (r! (n-r)!)$

To expand $(a+6)^4$ , substitute $a=a, b=6,$ and $n=4$ into the formula.

$(a+6)^4 = ((4),(0)) * a^4 * 6^0 +((4),(1)) * a^3 * 6^1 + ((4),(2)) * a^2 * 6^2 + ((4),(3)) * a^1 * 6^3 + ((4),(4)) * a^0 * 6^4$

$=1 * a^4 * 1 + 4 * a^3 * 6 + 6 * a^2 * 36 + 4 * a^1 * 216 + 1 * 1 * 1296$

$=a^4 + 24a^3 + 216a^2 + 864a + 1296$

How do you use the binomial theorem to expand and simplify the expression (a+6)^4(a+6)^4?