Expand Using the Binomial Theorem (7x+7)^2

${(7 x + 7)}^{2}$

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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

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

Apply the product rule to $7 x$ .

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

Multiply $7^{2}$ by ${(7)}^{0}$ by adding the exponents.

Move ${(7)}^{0}$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $0$ and $2$ .

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

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

$49 x^{2} + 2 \cdot {(7 x)}^{1} \cdot {(7)}^{1} + 1 \cdot {(7 x)}^{0} \cdot {(7)}^{2}$

Simplify.

$49 x^{2} + 2 \cdot (7 x) \cdot {(7)}^{1} + 1 \cdot {(7 x)}^{0} \cdot {(7)}^{2}$

Multiply $7$ by $2$ .

$49 x^{2} + 14 x \cdot {(7)}^{1} + 1 \cdot {(7 x)}^{0} \cdot {(7)}^{2}$

Evaluate the exponent.

$49 x^{2} + 14 x \cdot 7 + 1 \cdot {(7 x)}^{0} \cdot {(7)}^{2}$

Multiply $7$ by $14$ .

$49 x^{2} + 98 x + 1 \cdot {(7 x)}^{0} \cdot {(7)}^{2}$

Multiply ${(7 x)}^{0}$ by $1$ .

$49 x^{2} + 98 x + {(7 x)}^{0} \cdot {(7)}^{2}$

Apply the product rule to $7 x$ .

$49 x^{2} + 98 x + 7^{0} x^{0} \cdot {(7)}^{2}$

Multiply $7^{0}$ by ${(7)}^{2}$ by adding the exponents.

Move ${(7)}^{2}$ .

$49 x^{2} + 98 x + {(7)}^{2} \cdot 7^{0} x^{0}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$49 x^{2} + 98 x + 7^{2 + 0} x^{0}$

Add $2$ and $0$ .

$49 x^{2} + 98 x + 7^{2} x^{0}$

Simplify $7^{2} x^{0}$ .

$49 x^{2} + 98 x + 7^{2}$

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

$49 x^{2} + 98 x + 49$

Expand Using the Binomial Theorem (7x+7)^2

Related Questions

Support LCMGCF.com by disabling your adblocker.