Simplify (cd)^(2/3)(a^(1/4)d)^3

${(c d)}^{\frac{2}{3}} {(a^{\frac{1}{4}} d)}^{3}$

Step 1

Apply the product rule to $c d$ .

$c^{\frac{2}{3}} d^{\frac{2}{3}} {(a^{\frac{1}{4}} d)}^{3}$

Step 2

Apply the product rule to $a^{\frac{1}{4}} d$ .

$c^{\frac{2}{3}} d^{\frac{2}{3}} ({(a^{\frac{1}{4}})}^{3} d^{3})$

Step 3

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

Move $d^{3}$ .

$c^{\frac{2}{3}} (d^{3} d^{\frac{2}{3}}) {(a^{\frac{1}{4}})}^{3}$

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

$c^{\frac{2}{3}} d^{3 + \frac{2}{3}} {(a^{\frac{1}{4}})}^{3}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$c^{\frac{2}{3}} d^{3 \cdot \frac{3}{3} + \frac{2}{3}} {(a^{\frac{1}{4}})}^{3}$

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

$c^{\frac{2}{3}} d^{\frac{3 \cdot 3}{3} + \frac{2}{3}} {(a^{\frac{1}{4}})}^{3}$

Combine the numerators over the common denominator.

$c^{\frac{2}{3}} d^{\frac{3 \cdot 3 + 2}{3}} {(a^{\frac{1}{4}})}^{3}$

Simplify the numerator.

Multiply $3$ by $3$ .

$c^{\frac{2}{3}} d^{\frac{9 + 2}{3}} {(a^{\frac{1}{4}})}^{3}$

Add $9$ and $2$ .

$c^{\frac{2}{3}} d^{\frac{11}{3}} {(a^{\frac{1}{4}})}^{3}$

Step 4

Multiply the exponents in ${(a^{\frac{1}{4}})}^{3}$ .

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

$c^{\frac{2}{3}} d^{\frac{11}{3}} a^{\frac{1}{4} \cdot 3}$

Combine $\frac{1}{4}$ and $3$ .

$c^{\frac{2}{3}} d^{\frac{11}{3}} a^{\frac{3}{4}}$