Expand Using the Binomial Theorem (5y+7)^2

${(5 y + 7)}^{2}$

Step 1

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

$\sum_{k = 0}^{2} \frac{2!}{(2 - k)! k!} \cdot {(5 y)}^{2 - k} \cdot {(7)}^{k}$

Step 2

Expand the summation.

$\frac{2!}{(2 - 0)! 0!} \cdot {(5 y)}^{2 - 0} \cdot {(7)}^{0} + \frac{2!}{(2 - 1)! 1!} \cdot {(5 y)}^{2 - 1} \cdot {(7)}^{1} + \frac{2!}{(2 - 2)! 2!} \cdot {(5 y)}^{2 - 2} \cdot {(7)}^{2}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(5 y)}^{2} \cdot {(7)}^{0} + 2 \cdot {(5 y)}^{1} \cdot {(7)}^{1} + 1 \cdot {(5 y)}^{0} \cdot {(7)}^{2}$

Step 4

Simplify each term.

Multiply ${(5 y)}^{2}$ by $1$ .

${(5 y)}^{2} \cdot {(7)}^{0} + 2 \cdot {(5 y)}^{1} \cdot {(7)}^{1} + 1 \cdot {(5 y)}^{0} \cdot {(7)}^{2}$

Apply the product rule to $5 y$ .

$5^{2} y^{2} \cdot {(7)}^{0} + 2 \cdot {(5 y)}^{1} \cdot {(7)}^{1} + 1 \cdot {(5 y)}^{0} \cdot {(7)}^{2}$

Raise $5$ to the power of $2$ .

$25 y^{2} \cdot {(7)}^{0} + 2 \cdot {(5 y)}^{1} \cdot {(7)}^{1} + 1 \cdot {(5 y)}^{0} \cdot {(7)}^{2}$

Anything raised to $0$ is $1$ .

$25 y^{2} \cdot 1 + 2 \cdot {(5 y)}^{1} \cdot {(7)}^{1} + 1 \cdot {(5 y)}^{0} \cdot {(7)}^{2}$

Multiply $25$ by $1$ .

$25 y^{2} + 2 \cdot {(5 y)}^{1} \cdot {(7)}^{1} + 1 \cdot {(5 y)}^{0} \cdot {(7)}^{2}$

Simplify.

$25 y^{2} + 2 \cdot (5 y) \cdot {(7)}^{1} + 1 \cdot {(5 y)}^{0} \cdot {(7)}^{2}$

Multiply $5$ by $2$ .

$25 y^{2} + 10 y \cdot {(7)}^{1} + 1 \cdot {(5 y)}^{0} \cdot {(7)}^{2}$

Evaluate the exponent.

$25 y^{2} + 10 y \cdot 7 + 1 \cdot {(5 y)}^{0} \cdot {(7)}^{2}$

Multiply $7$ by $10$ .

$25 y^{2} + 70 y + 1 \cdot {(5 y)}^{0} \cdot {(7)}^{2}$

Multiply ${(5 y)}^{0}$ by $1$ .

$25 y^{2} + 70 y + {(5 y)}^{0} \cdot {(7)}^{2}$

Apply the product rule to $5 y$ .

$25 y^{2} + 70 y + 5^{0} y^{0} \cdot {(7)}^{2}$

Anything raised to $0$ is $1$ .

$25 y^{2} + 70 y + 1 y^{0} \cdot {(7)}^{2}$

Multiply $y^{0}$ by $1$ .

$25 y^{2} + 70 y + y^{0} \cdot {(7)}^{2}$

Anything raised to $0$ is $1$ .

$25 y^{2} + 70 y + 1 \cdot {(7)}^{2}$

Multiply ${(7)}^{2}$ by $1$ .

$25 y^{2} + 70 y + {(7)}^{2}$

Raise $7$ to the power of $2$ .

$25 y^{2} + 70 y + 49$

Expand Using the Binomial Theorem (5y+7)^2

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