Simplify (c^4*(a(-1/4)))^(3/2)

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

Step 1

Rewrite using the commutative property of multiplication.

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

Step 2

Combine $c^{4}$ and $\frac{1}{4}$ .

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

Step 3

Combine $a$ and $\frac{c^{4}}{4}$ .

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

Step 4

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

Apply the product rule to $- \frac{a c^{4}}{4}$ .

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

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

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

Apply the product rule to $a c^{4}$ .

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

Step 5

Rewrite ${(- 1)}^{\frac{3}{2}}$ as $\sqrt{{(- 1)}^{3}}$ .

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

Step 6

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

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

Step 7

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 8

Multiply the exponents in ${(c^{4})}^{\frac{3}{2}}$ .

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

$i \frac{a^{\frac{3}{2}} c^{2 \cdot 2 \frac{3}{2}}}{4^{\frac{3}{2}}}$

Rewrite the expression.

$i \frac{a^{\frac{3}{2}} c^{2 \cdot 3}}{4^{\frac{3}{2}}}$

Multiply $2$ by $3$ .

$i \frac{a^{\frac{3}{2}} c^{6}}{4^{\frac{3}{2}}}$

Step 9

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$i \frac{a^{\frac{3}{2}} c^{6}}{{(2^{2})}^{\frac{3}{2}}}$

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

$i \frac{a^{\frac{3}{2}} c^{6}}{2^{2 (\frac{3}{2})}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$i \frac{a^{\frac{3}{2}} c^{6}}{2^{2 (\frac{3}{2})}}$

Rewrite the expression.

$i \frac{a^{\frac{3}{2}} c^{6}}{2^{3}}$

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

$i \frac{a^{\frac{3}{2}} c^{6}}{8}$

Step 10

Combine fractions.

Combine $i$ and $\frac{a^{\frac{3}{2}} c^{6}}{8}$ .

$\frac{i (a^{\frac{3}{2}} c^{6})}{8}$

Reorder factors in $\frac{i a^{\frac{3}{2}} c^{6}}{8}$ .

$\frac{i c^{6} a^{\frac{3}{2}}}{8}$

Simplify (c^4*(a(-1/4)))^(3/2)

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