Math

QuestionCalculate (5+n)3(5+n)^{3}.

Studdy Solution

STEP 1

Assumptions1. We are asked to expand the expression (5+n)3(5+n)^{3}. . We will use the binomial theorem for expansion which states that (a+b)n=k=0n(nk)ankbk(a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k}b^{k}.

STEP 2

We can write (5+n)^ in the form of (a+b)n(a+b)^n where a=5a=5, b=nb=n and n=n=.

STEP 3

Now, we will use the binomial theorem to expand (5+n)3(5+n)^3.
(5+n)3=k=03(3k)53knk(5+n)^3 = \sum_{k=0}^{3} {3 \choose k}5^{3-k}n^{k}

STEP 4

We can expand this sum term by term.
(+n)3=(30)30n0+(31)31n1+(32)32n2+(33)33n3(+n)^3 = {3 \choose0}^{3-0}n^{0} + {3 \choose1}^{3-1}n^{1} + {3 \choose2}^{3-2}n^{2} + {3 \choose3}^{3-3}n^{3}

STEP 5

implify the binomial coefficients and the powers of5 and n.
(5+n)3=1531+352n+35n2+11n3(5+n)^3 =1 \cdot5^{3} \cdot1 +3 \cdot5^{2} \cdot n +3 \cdot5 \cdot n^{2} +1 \cdot1 \cdot n^{3}

STEP 6

Calculate the values.
(5+n)3=125+75n+15n2+n3(5+n)^3 =125 +75n +15n^{2} + n^{3}So, the expansion of (5+n)3(5+n)^3 is 125+75n+15n2+n3125 +75n +15n^{2} + n^{3}.

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