Find the Product (3m-4)^3(3m^5)

${(3 m - 4)}^{3} (3 m^{5})$

Step 1

Rewrite using the commutative property of multiplication.

$3 {(3 m - 4)}^{3} m^{5}$

Step 2

Use the Binomial Theorem.

$3 ({(3 m)}^{3} + 3 {(3 m)}^{2} \cdot - 4 + 3 (3 m) {(- 4)}^{2} + {(- 4)}^{3}) m^{5}$

Step 3

Simplify terms.

Simplify each term.

Apply the product rule to $3 m$ .

$3 (3^{3} m^{3} + 3 {(3 m)}^{2} \cdot - 4 + 3 (3 m) {(- 4)}^{2} + {(- 4)}^{3}) m^{5}$

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

$3 (27 m^{3} + 3 {(3 m)}^{2} \cdot - 4 + 3 (3 m) {(- 4)}^{2} + {(- 4)}^{3}) m^{5}$

Apply the product rule to $3 m$ .

$3 (27 m^{3} + 3 (3^{2} m^{2}) \cdot - 4 + 3 (3 m) {(- 4)}^{2} + {(- 4)}^{3}) m^{5}$

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

Move $3^{2}$ .

$3 (27 m^{3} + 3^{2} \cdot 3 m^{2} \cdot - 4 + 3 (3 m) {(- 4)}^{2} + {(- 4)}^{3}) m^{5}$

Multiply $3^{2}$ by $3$ .

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

$3 (27 m^{3} + 3^{2} \cdot 3^{1} m^{2} \cdot - 4 + 3 (3 m) {(- 4)}^{2} + {(- 4)}^{3}) m^{5}$

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

$3 (27 m^{3} + 3^{2 + 1} m^{2} \cdot - 4 + 3 (3 m) {(- 4)}^{2} + {(- 4)}^{3}) m^{5}$

Add $2$ and $1$ .

$3 (27 m^{3} + 3^{3} m^{2} \cdot - 4 + 3 (3 m) {(- 4)}^{2} + {(- 4)}^{3}) m^{5}$

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

$3 (27 m^{3} + 27 m^{2} \cdot - 4 + 3 (3 m) {(- 4)}^{2} + {(- 4)}^{3}) m^{5}$

Multiply $- 4$ by $27$ .

$3 (27 m^{3} - 108 m^{2} + 3 (3 m) {(- 4)}^{2} + {(- 4)}^{3}) m^{5}$

Multiply $3$ by $3$ .

$3 (27 m^{3} - 108 m^{2} + 9 m {(- 4)}^{2} + {(- 4)}^{3}) m^{5}$

Raise $- 4$ to the power of $2$ .

$3 (27 m^{3} - 108 m^{2} + 9 m \cdot 16 + {(- 4)}^{3}) m^{5}$

Multiply $16$ by $9$ .

$3 (27 m^{3} - 108 m^{2} + 144 m + {(- 4)}^{3}) m^{5}$

Raise $- 4$ to the power of $3$ .

$3 (27 m^{3} - 108 m^{2} + 144 m - 64) m^{5}$

Apply the distributive property.

$(3 (27 m^{3}) + 3 (- 108 m^{2}) + 3 (144 m) + 3 \cdot - 64) m^{5}$

Step 4

Simplify.

Multiply $27$ by $3$ .

$(81 m^{3} + 3 (- 108 m^{2}) + 3 (144 m) + 3 \cdot - 64) m^{5}$

Multiply $- 108$ by $3$ .

$(81 m^{3} - 324 m^{2} + 3 (144 m) + 3 \cdot - 64) m^{5}$

Multiply $144$ by $3$ .

$(81 m^{3} - 324 m^{2} + 432 m + 3 \cdot - 64) m^{5}$

Multiply $3$ by $- 64$ .

$(81 m^{3} - 324 m^{2} + 432 m - 192) m^{5}$

Step 5

Apply the distributive property.

$81 m^{3} m^{5} - 324 m^{2} m^{5} + 432 m \cdot m^{5} - 192 m^{5}$

Step 6

Simplify.

Multiply $m^{3}$ by $m^{5}$ by adding the exponents.

Move $m^{5}$ .

$81 (m^{5} m^{3}) - 324 m^{2} m^{5} + 432 m \cdot m^{5} - 192 m^{5}$

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

$81 m^{5 + 3} - 324 m^{2} m^{5} + 432 m \cdot m^{5} - 192 m^{5}$

Add $5$ and $3$ .

$81 m^{8} - 324 m^{2} m^{5} + 432 m \cdot m^{5} - 192 m^{5}$

Multiply $m^{2}$ by $m^{5}$ by adding the exponents.

Move $m^{5}$ .

$81 m^{8} - 324 (m^{5} m^{2}) + 432 m \cdot m^{5} - 192 m^{5}$

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

$81 m^{8} - 324 m^{5 + 2} + 432 m \cdot m^{5} - 192 m^{5}$

Add $5$ and $2$ .

$81 m^{8} - 324 m^{7} + 432 m \cdot m^{5} - 192 m^{5}$

Multiply $m$ by $m^{5}$ by adding the exponents.

Move $m^{5}$ .

$81 m^{8} - 324 m^{7} + 432 (m^{5} m) - 192 m^{5}$

Multiply $m^{5}$ by $m$ .

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

$81 m^{8} - 324 m^{7} + 432 (m^{5} m^{1}) - 192 m^{5}$

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

$81 m^{8} - 324 m^{7} + 432 m^{5 + 1} - 192 m^{5}$

Add $5$ and $1$ .

$81 m^{8} - 324 m^{7} + 432 m^{6} - 192 m^{5}$

Find the Product (3m-4)^3(3m^5)

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