Expand Using the Binomial Theorem (m+1/m)^6

${(m + \frac{1}{m})}^{6}$

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{6} \frac{6!}{(6 - k)! k!} \cdot {(m)}^{6 - k} \cdot {(\frac{1}{m})}^{k}$

Step 2

Expand the summation.

$\frac{6!}{(6 - 0)! 0!} {(m)}^{6 - 0} \cdot {(\frac{1}{m})}^{0} + \frac{6!}{(6 - 1)! 1!} {(m)}^{6 - 1} \cdot {(\frac{1}{m})}^{1} + \dots + \frac{6!}{(6 - 6)! 6!} {(m)}^{6 - 6} \cdot {(\frac{1}{m})}^{6}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(m)}^{6} \cdot {(\frac{1}{m})}^{0} + 6 \cdot {(m)}^{5} \cdot {(\frac{1}{m})}^{1} + \dots + 1 \cdot {(m)}^{0} \cdot {(\frac{1}{m})}^{6}$

Step 4

Simplify the polynomial result.

$m^{6} + 6 m^{4} + 15 m^{2} + 20 + \frac{15}{m^{2}} + \frac{6}{m^{4}} + \frac{1}{m^{6}}$

Expand Using the Binomial Theorem (m+1/m)^6

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