Simplify ((12p^(3/2))/(15p))^4

${(\frac{12 p^{\frac{3}{2}}}{15 p})}^{4}$

Step 1

Move $p^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

${(\frac{12 p^{\frac{3}{2}} p^{- 1}}{15})}^{4}$

Step 2

Multiply $p^{\frac{3}{2}}$ by $p^{- 1}$ by adding the exponents.

Move $p^{- 1}$ .

${(\frac{12 (p^{- 1} p^{\frac{3}{2}})}{15})}^{4}$

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

${(\frac{12 p^{- 1 + \frac{3}{2}}}{15})}^{4}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(\frac{12 p^{- 1 \cdot \frac{2}{2} + \frac{3}{2}}}{15})}^{4}$

Combine $- 1$ and $\frac{2}{2}$ .

${(\frac{12 p^{\frac{- 1 \cdot 2}{2} + \frac{3}{2}}}{15})}^{4}$

Combine the numerators over the common denominator.

${(\frac{12 p^{\frac{- 1 \cdot 2 + 3}{2}}}{15})}^{4}$

Simplify the numerator.

Multiply $- 1$ by $2$ .

${(\frac{12 p^{\frac{- 2 + 3}{2}}}{15})}^{4}$

Add $- 2$ and $3$ .

${(\frac{12 p^{\frac{1}{2}}}{15})}^{4}$

Step 3

Factor $3$ out of $12 p^{\frac{1}{2}}$ .

${(\frac{3 (4 p^{\frac{1}{2}})}{15})}^{4}$

Step 4

Cancel the common factors.

Factor $3$ out of $15$ .

${(\frac{3 (4 p^{\frac{1}{2}})}{3 (5)})}^{4}$

Cancel the common factor.

${(\frac{3 (4 p^{\frac{1}{2}})}{3 \cdot 5})}^{4}$

Rewrite the expression.

${(\frac{4 p^{\frac{1}{2}}}{5})}^{4}$

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{4 p^{\frac{1}{2}}}{5}$ .

$\frac{{(4 p^{\frac{1}{2}})}^{4}}{5^{4}}$

Apply the product rule to $4 p^{\frac{1}{2}}$ .

$\frac{4^{4} {(p^{\frac{1}{2}})}^{4}}{5^{4}}$

Step 6

Simplify the numerator.

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

$\frac{256 {(p^{\frac{1}{2}})}^{4}}{5^{4}}$

Multiply the exponents in ${(p^{\frac{1}{2}})}^{4}$ .

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

$\frac{256 p^{\frac{1}{2} \cdot 4}}{5^{4}}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$\frac{256 p^{\frac{1}{2} \cdot (2 (2))}}{5^{4}}$

Cancel the common factor.

$\frac{256 p^{\frac{1}{2} \cdot (2 \cdot 2)}}{5^{4}}$

Rewrite the expression.

$\frac{256 p^{2}}{5^{4}}$

Step 7

Raise $5$ to the power of $4$ .

$\frac{256 p^{2}}{625}$

Simplify ((12p^(3/2))/(15p))^4

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