Expand the Trigonometric Expression (x^2+9y)^3

${(x^{2} + 9 y)}^{3}$

Step 1

Use the Binomial Theorem.

${(x^{2})}^{3} + 3 {(x^{2})}^{2} (9 y) + 3 x^{2} {(9 y)}^{2} + {(9 y)}^{3}$

Step 2

Simplify each term.

Multiply the exponents in ${(x^{2})}^{3}$ .

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

$x^{2 \cdot 3} + 3 {(x^{2})}^{2} (9 y) + 3 x^{2} {(9 y)}^{2} + {(9 y)}^{3}$

Multiply $2$ by $3$ .

$x^{6} + 3 {(x^{2})}^{2} (9 y) + 3 x^{2} {(9 y)}^{2} + {(9 y)}^{3}$

Rewrite using the commutative property of multiplication.

$x^{6} + 3 \cdot 9 {(x^{2})}^{2} y + 3 x^{2} {(9 y)}^{2} + {(9 y)}^{3}$

Multiply $3$ by $9$ .

$x^{6} + 27 {(x^{2})}^{2} y + 3 x^{2} {(9 y)}^{2} + {(9 y)}^{3}$

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

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

$x^{6} + 27 x^{2 \cdot 2} y + 3 x^{2} {(9 y)}^{2} + {(9 y)}^{3}$

Multiply $2$ by $2$ .

$x^{6} + 27 x^{4} y + 3 x^{2} {(9 y)}^{2} + {(9 y)}^{3}$

Apply the product rule to $9 y$ .

$x^{6} + 27 x^{4} y + 3 x^{2} (9^{2} y^{2}) + {(9 y)}^{3}$

Rewrite using the commutative property of multiplication.

$x^{6} + 27 x^{4} y + 3 \cdot 9^{2} x^{2} y^{2} + {(9 y)}^{3}$

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

$x^{6} + 27 x^{4} y + 3 \cdot 81 x^{2} y^{2} + {(9 y)}^{3}$

Multiply $3$ by $81$ .

$x^{6} + 27 x^{4} y + 243 x^{2} y^{2} + {(9 y)}^{3}$

Apply the product rule to $9 y$ .

$x^{6} + 27 x^{4} y + 243 x^{2} y^{2} + 9^{3} y^{3}$

Raise $9$ to the power of $3$ .

$x^{6} + 27 x^{4} y + 243 x^{2} y^{2} + 729 y^{3}$