Simplify (5t^2)^3(3t)^2

${(5 t^{2})}^{3} {(3 t)}^{2}$

Step 1

Apply the product rule to $5 t^{2}$ .

$5^{3} {(t^{2})}^{3} {(3 t)}^{2}$

Step 2

Raise $5$ to the power of $3$ .

$125 {(t^{2})}^{3} {(3 t)}^{2}$

Step 3

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

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

$125 t^{2 \cdot 3} {(3 t)}^{2}$

Multiply $2$ by $3$ .

$125 t^{6} {(3 t)}^{2}$

Step 4

Apply the product rule to $3 t$ .

$125 t^{6} (3^{2} t^{2})$

Step 5

Rewrite using the commutative property of multiplication.

$125 \cdot 3^{2} t^{6} t^{2}$

Step 6

Multiply $t^{6}$ by $t^{2}$ by adding the exponents.

Move $t^{2}$ .

$125 \cdot 3^{2} (t^{2} t^{6})$

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

$125 \cdot 3^{2} t^{2 + 6}$

Add $2$ and $6$ .

$125 \cdot 3^{2} t^{8}$

Step 7

Raise $3$ to the power of $2$ .

$125 \cdot 9 t^{8}$

Step 8

Multiply $125$ by $9$ .

$1125 t^{8}$