Expand Using the Binomial Theorem (-9x^2-10y)^2

${(- 9 x^{2} - 10 y)}^{2}$

Step 1

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

$\sum_{k = 0}^{2} \frac{2!}{(2 - k)! k!} \cdot {(- 9 x^{2})}^{2 - k} \cdot {(- 10 y)}^{k}$

Step 2

Expand the summation.

$\frac{2!}{(2 - 0)! 0!} \cdot {(- 9 x^{2})}^{2 - 0} \cdot {(- 10 y)}^{0} + \frac{2!}{(2 - 1)! 1!} \cdot {(- 9 x^{2})}^{2 - 1} \cdot {(- 10 y)}^{1} + \frac{2!}{(2 - 2)! 2!} \cdot {(- 9 x^{2})}^{2 - 2} \cdot {(- 10 y)}^{2}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(- 9 x^{2})}^{2} \cdot {(- 10 y)}^{0} + 2 \cdot {(- 9 x^{2})}^{1} \cdot {(- 10 y)}^{1} + 1 \cdot {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Step 4

Simplify each term.

Multiply ${(- 9 x^{2})}^{2}$ by $1$ .

${(- 9 x^{2})}^{2} \cdot {(- 10 y)}^{0} + 2 \cdot {(- 9 x^{2})}^{1} \cdot {(- 10 y)}^{1} + 1 \cdot {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Apply the product rule to $- 9 x^{2}$ .

${(- 9)}^{2} {(x^{2})}^{2} \cdot {(- 10 y)}^{0} + 2 \cdot {(- 9 x^{2})}^{1} \cdot {(- 10 y)}^{1} + 1 \cdot {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Raise $- 9$ to the power of $2$ .

$81 {(x^{2})}^{2} \cdot {(- 10 y)}^{0} + 2 \cdot {(- 9 x^{2})}^{1} \cdot {(- 10 y)}^{1} + 1 \cdot {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Multiply the exponents in ${(x^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$81 x^{2 \cdot 2} \cdot {(- 10 y)}^{0} + 2 \cdot {(- 9 x^{2})}^{1} \cdot {(- 10 y)}^{1} + 1 \cdot {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Multiply $2$ by $2$ .

$81 x^{4} \cdot {(- 10 y)}^{0} + 2 \cdot {(- 9 x^{2})}^{1} \cdot {(- 10 y)}^{1} + 1 \cdot {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Apply the product rule to $- 10 y$ .

$81 x^{4} \cdot ({(- 10)}^{0} y^{0}) + 2 \cdot {(- 9 x^{2})}^{1} \cdot {(- 10 y)}^{1} + 1 \cdot {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Rewrite using the commutative property of multiplication.

$81 \cdot {(- 10)}^{0} x^{4} y^{0} + 2 \cdot {(- 9 x^{2})}^{1} \cdot {(- 10 y)}^{1} + 1 \cdot {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Anything raised to $0$ is $1$ .

$81 \cdot 1 x^{4} y^{0} + 2 \cdot {(- 9 x^{2})}^{1} \cdot {(- 10 y)}^{1} + 1 \cdot {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Multiply $81$ by $1$ .

$81 x^{4} y^{0} + 2 \cdot {(- 9 x^{2})}^{1} \cdot {(- 10 y)}^{1} + 1 \cdot {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Anything raised to $0$ is $1$ .

$81 x^{4} \cdot 1 + 2 \cdot {(- 9 x^{2})}^{1} \cdot {(- 10 y)}^{1} + 1 \cdot {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Multiply $81$ by $1$ .

$81 x^{4} + 2 \cdot {(- 9 x^{2})}^{1} \cdot {(- 10 y)}^{1} + 1 \cdot {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Simplify.

$81 x^{4} + 2 \cdot (- 9 x^{2}) \cdot {(- 10 y)}^{1} + 1 \cdot {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Multiply $- 9$ by $2$ .

$81 x^{4} - 18 x^{2} \cdot {(- 10 y)}^{1} + 1 \cdot {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Simplify.

$81 x^{4} - 18 x^{2} \cdot (- 10 y) + 1 \cdot {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Rewrite using the commutative property of multiplication.

$81 x^{4} - 18 \cdot - 10 x^{2} y + 1 \cdot {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Multiply $- 18$ by $- 10$ .

$81 x^{4} + 180 x^{2} y + 1 \cdot {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Multiply ${(- 9 x^{2})}^{0}$ by $1$ .

$81 x^{4} + 180 x^{2} y + {(- 9 x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Apply the product rule to $- 9 x^{2}$ .

$81 x^{4} + 180 x^{2} y + {(- 9)}^{0} {(x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Anything raised to $0$ is $1$ .

$81 x^{4} + 180 x^{2} y + 1 {(x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Multiply ${(x^{2})}^{0}$ by $1$ .

$81 x^{4} + 180 x^{2} y + {(x^{2})}^{0} \cdot {(- 10 y)}^{2}$

Multiply the exponents in ${(x^{2})}^{0}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$81 x^{4} + 180 x^{2} y + x^{2 \cdot 0} \cdot {(- 10 y)}^{2}$

Multiply $2$ by $0$ .

$81 x^{4} + 180 x^{2} y + x^{0} \cdot {(- 10 y)}^{2}$

Anything raised to $0$ is $1$ .

$81 x^{4} + 180 x^{2} y + 1 \cdot {(- 10 y)}^{2}$

Multiply ${(- 10 y)}^{2}$ by $1$ .

$81 x^{4} + 180 x^{2} y + {(- 10 y)}^{2}$

Apply the product rule to $- 10 y$ .

$81 x^{4} + 180 x^{2} y + {(- 10)}^{2} y^{2}$

Raise $- 10$ to the power of $2$ .

$81 x^{4} + 180 x^{2} y + 100 y^{2}$

Expand Using the Binomial Theorem (-9x^2-10y)^2

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