Expand the Trigonometric Expression (x^2-y)^3

${(x^{2} - y)}^{3}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

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

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

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

Multiply $2$ by $3$ .

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

Rewrite using the commutative property of multiplication.

$x^{6} + 3 \cdot - 1 {(x^{2})}^{2} y + 3 x^{2} {(- y)}^{2} + {(- y)}^{3}$

Multiply $3$ by $- 1$ .

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

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

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

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

Multiply $2$ by $2$ .

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

Apply the product rule to $- y$ .

$x^{6} - 3 x^{4} y + 3 x^{2} ({(- 1)}^{2} y^{2}) + {(- y)}^{3}$

Rewrite using the commutative property of multiplication.

$x^{6} - 3 x^{4} y + 3 \cdot {(- 1)}^{2} x^{2} y^{2} + {(- y)}^{3}$

Raise $- 1$ to the power of $2$ .

$x^{6} - 3 x^{4} y + 3 \cdot 1 x^{2} y^{2} + {(- y)}^{3}$

Multiply $3$ by $1$ .

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

Apply the product rule to $- y$ .

$x^{6} - 3 x^{4} y + 3 x^{2} y^{2} + {(- 1)}^{3} y^{3}$

Raise $- 1$ to the power of $3$ .

$x^{6} - 3 x^{4} y + 3 x^{2} y^{2} - y^{3}$