Simplify ((-8x^6)/(y^(1/2)z^(3/2)))^(2/3)

${(\frac{- 8 x^{6}}{y^{\frac{1}{2}} z^{\frac{3}{2}}})}^{\frac{2}{3}}$

Step 1

Move the negative in front of the fraction.

${(- \frac{(8) x^{6}}{y^{\frac{1}{2}} z^{\frac{3}{2}}})}^{\frac{2}{3}}$

Step 2

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

Apply the product rule to $- \frac{8 x^{6}}{y^{\frac{1}{2}} z^{\frac{3}{2}}}$ .

${(- 1)}^{\frac{2}{3}} {(\frac{8 x^{6}}{y^{\frac{1}{2}} z^{\frac{3}{2}}})}^{\frac{2}{3}}$

Apply the product rule to $\frac{8 x^{6}}{y^{\frac{1}{2}} z^{\frac{3}{2}}}$ .

${(- 1)}^{\frac{2}{3}} \frac{{(8 x^{6})}^{\frac{2}{3}}}{{(y^{\frac{1}{2}} z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Apply the product rule to $8 x^{6}$ .

${(- 1)}^{\frac{2}{3}} \frac{8^{\frac{2}{3}} {(x^{6})}^{\frac{2}{3}}}{{(y^{\frac{1}{2}} z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Apply the product rule to $y^{\frac{1}{2}} z^{\frac{3}{2}}$ .

${(- 1)}^{\frac{2}{3}} \frac{8^{\frac{2}{3}} {(x^{6})}^{\frac{2}{3}}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Step 3

Reduce the expression by cancelling the common factors.

Rewrite $- 1$ as ${(- 1)}^{3}$ .

${({(- 1)}^{3})}^{\frac{2}{3}} \frac{8^{\frac{2}{3}} {(x^{6})}^{\frac{2}{3}}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

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

${(- 1)}^{3 (\frac{2}{3})} \frac{8^{\frac{2}{3}} {(x^{6})}^{\frac{2}{3}}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(- 1)}^{3 (\frac{2}{3})} \frac{8^{\frac{2}{3}} {(x^{6})}^{\frac{2}{3}}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Rewrite the expression.

${(- 1)}^{2} \frac{8^{\frac{2}{3}} {(x^{6})}^{\frac{2}{3}}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Simplify the expression.

Raise $- 1$ to the power of $2$ .

$1 \frac{8^{\frac{2}{3}} {(x^{6})}^{\frac{2}{3}}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Multiply $\frac{8^{\frac{2}{3}} {(x^{6})}^{\frac{2}{3}}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$ by $1$ .

$\frac{8^{\frac{2}{3}} {(x^{6})}^{\frac{2}{3}}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Step 4

Simplify the numerator.

Rewrite $8$ as $2^{3}$ .

$\frac{{(2^{3})}^{\frac{2}{3}} {(x^{6})}^{\frac{2}{3}}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

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

$\frac{2^{3 (\frac{2}{3})} {(x^{6})}^{\frac{2}{3}}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{2^{3 (\frac{2}{3})} {(x^{6})}^{\frac{2}{3}}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Rewrite the expression.

$\frac{2^{2} {(x^{6})}^{\frac{2}{3}}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

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

$\frac{4 {(x^{6})}^{\frac{2}{3}}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Multiply the exponents in ${(x^{6})}^{\frac{2}{3}}$ .

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

$\frac{4 x^{6 (\frac{2}{3})}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$\frac{4 x^{3 (2) \frac{2}{3}}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Cancel the common factor.

$\frac{4 x^{3 \cdot 2 \frac{2}{3}}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Rewrite the expression.

$\frac{4 x^{2 \cdot 2}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Multiply $2$ by $2$ .

$\frac{4 x^{4}}{{(y^{\frac{1}{2}})}^{\frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Step 5

Simplify the denominator.

Multiply the exponents in ${(y^{\frac{1}{2}})}^{\frac{2}{3}}$ .

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

$\frac{4 x^{4}}{y^{\frac{1}{2} \cdot \frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 x^{4}}{y^{\frac{1}{2} \cdot \frac{2}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

Rewrite the expression.

$\frac{4 x^{4}}{y^{\frac{1}{3}} {(z^{\frac{3}{2}})}^{\frac{2}{3}}}$

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

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

$\frac{4 x^{4}}{y^{\frac{1}{3}} z^{\frac{3}{2} \cdot \frac{2}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{4 x^{4}}{y^{\frac{1}{3}} z^{\frac{3}{2} \cdot \frac{2}{3}}}$

Rewrite the expression.

$\frac{4 x^{4}}{y^{\frac{1}{3}} z^{\frac{1}{2} \cdot 2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 x^{4}}{y^{\frac{1}{3}} z^{\frac{1}{2} \cdot 2}}$

Rewrite the expression.

$\frac{4 x^{4}}{y^{\frac{1}{3}} z^{1}}$

Simplify.

$\frac{4 x^{4}}{y^{\frac{1}{3}} z}$

Step 6

Reorder factors in $\frac{4 x^{4}}{y^{\frac{1}{3}} z}$ .

$\frac{4 x^{4}}{z y^{\frac{1}{3}}}$

Simplify ((-8x^6)/(y^(1/2)z^(3/2)))^(2/3)

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