Expand the Trigonometric Expression -4t^3(3r+2rt-5t^2)

$- 4 t^{3} (3 r + 2 r t - 5 t^{2})$

Step 1

Apply the distributive property.

$- 4 t^{3} (3 r) - 4 t^{3} (2 r t) - 4 t^{3} (- 5 t^{2})$

Step 2

Simplify.

Rewrite using the commutative property of multiplication.

$- 4 \cdot 3 t^{3} r - 4 t^{3} (2 r t) - 4 t^{3} (- 5 t^{2})$

Multiply $t^{3}$ by $t$ by adding the exponents.

Move $t$ .

$- 4 \cdot 3 t^{3} r - 4 (t \cdot t^{3}) (2 r) - 4 t^{3} (- 5 t^{2})$

Multiply $t$ by $t^{3}$ .

Raise $t$ to the power of $1$ .

$- 4 \cdot 3 t^{3} r - 4 (t^{1} t^{3}) (2 r) - 4 t^{3} (- 5 t^{2})$

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

$- 4 \cdot 3 t^{3} r - 4 t^{1 + 3} (2 r) - 4 t^{3} (- 5 t^{2})$

Add $1$ and $3$ .

$- 4 \cdot 3 t^{3} r - 4 t^{4} (2 r) - 4 t^{3} (- 5 t^{2})$

Rewrite using the commutative property of multiplication.

$- 4 \cdot 3 t^{3} r - 4 t^{4} (2 r) - 4 \cdot - 5 t^{3} t^{2}$

Step 3

Simplify each term.

Multiply $- 4$ by $3$ .

$- 12 t^{3} r - 4 t^{4} (2 r) - 4 \cdot - 5 t^{3} t^{2}$

Rewrite using the commutative property of multiplication.

$- 12 t^{3} r - 4 \cdot 2 t^{4} r - 4 \cdot - 5 t^{3} t^{2}$

Multiply $- 4$ by $2$ .

$- 12 t^{3} r - 8 t^{4} r - 4 \cdot - 5 t^{3} t^{2}$

Multiply $t^{3}$ by $t^{2}$ by adding the exponents.

Move $t^{2}$ .

$- 12 t^{3} r - 8 t^{4} r - 4 \cdot - 5 (t^{2} t^{3})$

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

$- 12 t^{3} r - 8 t^{4} r - 4 \cdot - 5 t^{2 + 3}$

Add $2$ and $3$ .

$- 12 t^{3} r - 8 t^{4} r - 4 \cdot - 5 t^{5}$

Multiply $- 4$ by $- 5$ .

$- 12 t^{3} r - 8 t^{4} r + 20 t^{5}$