Simplify (3/4z)(-8z^3)^2

$(\frac{3}{4} z) {(- 8 z^{3})}^{2}$

Step 1

Combine $\frac{3}{4}$ and $z$ .

$\frac{3 z}{4} {(- 8 z^{3})}^{2}$

Step 2

Simplify the expression.

Apply the product rule to $- 8 z^{3}$ .

$\frac{3 z}{4} ({(- 8)}^{2} {(z^{3})}^{2})$

Rewrite using the commutative property of multiplication.

${(- 8)}^{2} \frac{3 z}{4} {(z^{3})}^{2}$

Raise $- 8$ to the power of $2$ .

$64 \frac{3 z}{4} {(z^{3})}^{2}$

Step 3

Cancel the common factor of $4$ .

Factor $4$ out of $64$ .

$4 (16) \frac{3 z}{4} {(z^{3})}^{2}$

Cancel the common factor.

$4 \cdot 16 \frac{3 z}{4} {(z^{3})}^{2}$

Rewrite the expression.

$16 (3 z) {(z^{3})}^{2}$

Step 4

Simplify the expression.

Multiply $3$ by $16$ .

$48 z {(z^{3})}^{2}$

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

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

$48 z \cdot z^{3 \cdot 2}$

Multiply $3$ by $2$ .

$48 z \cdot z^{6}$

Step 5

Multiply $z$ by $z^{6}$ by adding the exponents.

Move $z^{6}$ .

$48 (z^{6} z)$

Multiply $z^{6}$ by $z$ .

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

$48 (z^{6} z^{1})$

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

$48 z^{6 + 1}$

Add $6$ and $1$ .

$48 z^{7}$