Expand Using the Binomial Theorem (7x+5y)^7

${(7 x + 5 y)}^{7}$

Step 1

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

$\sum_{k = 0}^{7} \frac{7!}{(7 - k)! k!} \cdot {(7 x)}^{7 - k} \cdot {(5 y)}^{k}$

Step 2

Expand the summation.

$\frac{7!}{(7 - 0)! 0!} \cdot {(7 x)}^{7 - 0} \cdot {(5 y)}^{0} + \frac{7!}{(7 - 1)! 1!} \cdot {(7 x)}^{7 - 1} \cdot {(5 y)}^{1} + \dots + \frac{7!}{(7 - 7)! 7!} \cdot {(7 x)}^{7 - 7} \cdot {(5 y)}^{7}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(7 x)}^{7} \cdot {(5 y)}^{0} + 7 \cdot {(7 x)}^{6} \cdot {(5 y)}^{1} + \dots + 1 \cdot {(7 x)}^{0} \cdot {(5 y)}^{7}$

Step 4

Simplify the polynomial result.

$823543 x^{7} + 4117715 x^{6} y + 8823675 x^{5} y^{2} + 10504375 x^{4} y^{3} + 7503125 x^{3} y^{4} + 3215625 x^{2} y^{5} + 765625 x y^{6} + 78125 y^{7}$

Expand Using the Binomial Theorem (7x+5y)^7

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