Simplify the Radical Expression ( square root of z^9)/( fourth root of z^3)

$\frac{\sqrt{z^{9}}}{\sqrt[4]{z^{3}}}$

Simplify the numerator.

Rewrite $z^{9}$ as ${(z^{4})}^{2} z$ .

Factor out $z^{8}$ .

$\frac{\sqrt{z^{8} z}}{\sqrt[4]{z^{3}}}$

Rewrite $z^{8}$ as ${(z^{4})}^{2}$ .

$\frac{\sqrt{{(z^{4})}^{2} z}}{\sqrt[4]{z^{3}}}$

Pull terms out from under the radical.

$\frac{| z^{4} | \sqrt{z}}{\sqrt[4]{z^{3}}}$

Remove non-negative terms from the absolute value.

$\frac{z^{4} \sqrt{z}}{\sqrt[4]{z^{3}}}$

Multiply $\frac{z^{4} \sqrt{z}}{\sqrt[4]{z^{3}}}$ by $\frac{{\sqrt[4]{z^{3}}}^{3}}{{\sqrt[4]{z^{3}}}^{3}}$ .

$\frac{z^{4} \sqrt{z}}{\sqrt[4]{z^{3}}} \cdot \frac{{\sqrt[4]{z^{3}}}^{3}}{{\sqrt[4]{z^{3}}}^{3}}$

Combine and simplify the denominator.

Multiply $\frac{z^{4} \sqrt{z}}{\sqrt[4]{z^{3}}}$ by $\frac{{\sqrt[4]{z^{3}}}^{3}}{{\sqrt[4]{z^{3}}}^{3}}$ .

$\frac{z^{4} \sqrt{z} {\sqrt[4]{z^{3}}}^{3}}{\sqrt[4]{z^{3}} {\sqrt[4]{z^{3}}}^{3}}$

Raise $\sqrt[4]{z^{3}}$ to the power of $1$ .

$\frac{z^{4} \sqrt{z} {\sqrt[4]{z^{3}}}^{3}}{{\sqrt[4]{z^{3}}}^{1} {\sqrt[4]{z^{3}}}^{3}}$

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

$\frac{z^{4} \sqrt{z} {\sqrt[4]{z^{3}}}^{3}}{{\sqrt[4]{z^{3}}}^{1 + 3}}$

Add $1$ and $3$ .

$\frac{z^{4} \sqrt{z} {\sqrt[4]{z^{3}}}^{3}}{{\sqrt[4]{z^{3}}}^{4}}$

Rewrite ${\sqrt[4]{z^{3}}}^{4}$ as $z^{3}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{z^{3}}$ as $z^{\frac{3}{4}}$ .

$\frac{z^{4} \sqrt{z} {\sqrt[4]{z^{3}}}^{3}}{{(z^{\frac{3}{4}})}^{4}}$

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

$\frac{z^{4} \sqrt{z} {\sqrt[4]{z^{3}}}^{3}}{z^{\frac{3}{4} \cdot 4}}$

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

$\frac{z^{4} \sqrt{z} {\sqrt[4]{z^{3}}}^{3}}{z^{\frac{3 \cdot 4}{4}}}$

Multiply $3$ by $4$ .

$\frac{z^{4} \sqrt{z} {\sqrt[4]{z^{3}}}^{3}}{z^{\frac{12}{4}}}$

Cancel the common factor of $12$ and $4$ .

Factor $4$ out of $12$ .

$\frac{z^{4} \sqrt{z} {\sqrt[4]{z^{3}}}^{3}}{z^{\frac{4 \cdot 3}{4}}}$

Cancel the common factors.

Factor $4$ out of $4$ .

$\frac{z^{4} \sqrt{z} {\sqrt[4]{z^{3}}}^{3}}{z^{\frac{4 \cdot 3}{4 (1)}}}$

Cancel the common factor.

$\frac{z^{4} \sqrt{z} {\sqrt[4]{z^{3}}}^{3}}{z^{\frac{4 \cdot 3}{4 \cdot 1}}}$

Rewrite the expression.

$\frac{z^{4} \sqrt{z} {\sqrt[4]{z^{3}}}^{3}}{z^{\frac{3}{1}}}$

Divide $3$ by $1$ .

$\frac{z^{4} \sqrt{z} {\sqrt[4]{z^{3}}}^{3}}{z^{3}}$

Cancel the common factor of $z^{4}$ and $z^{3}$ .

Factor $z^{3}$ out of $z^{4} \sqrt{z} {\sqrt[4]{z^{3}}}^{3}$ .

$\frac{z^{3} (z \sqrt{z} {\sqrt[4]{z^{3}}}^{3})}{z^{3}}$

Cancel the common factors.

Multiply by $1$ .

$\frac{z^{3} (z \sqrt{z} {\sqrt[4]{z^{3}}}^{3})}{z^{3} \cdot 1}$

Cancel the common factor.

$\frac{z^{3} (z \sqrt{z} {\sqrt[4]{z^{3}}}^{3})}{z^{3} \cdot 1}$

Rewrite the expression.

$\frac{z \sqrt{z} {\sqrt[4]{z^{3}}}^{3}}{1}$

Divide $z \sqrt{z} {\sqrt[4]{z^{3}}}^{3}$ by $1$ .

$z \sqrt{z} {\sqrt[4]{z^{3}}}^{3}$

Rewrite ${\sqrt[4]{z^{3}}}^{3}$ as ${({(z^{3})}^{3})}^{\frac{1}{4}}$ .

$z \sqrt{z} \sqrt[4]{{(z^{3})}^{3}}$

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

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

$z \sqrt{z} \sqrt[4]{z^{3 \cdot 3}}$

Multiply $3$ by $3$ .

$z \sqrt{z} \sqrt[4]{z^{9}}$

Rewrite $z^{9}$ as ${(z^{2})}^{4} z$ .

Factor out $z^{8}$ .

$z \sqrt{z} \sqrt[4]{z^{8} z}$

Rewrite $z^{8}$ as ${(z^{2})}^{4}$ .

$z \sqrt{z} \sqrt[4]{{(z^{2})}^{4} z}$

Pull terms out from under the radical.

$z \sqrt{z} (| z^{2} | \sqrt[4]{z})$

Remove non-negative terms from the absolute value.

$z \sqrt{z} (z^{2} \sqrt[4]{z})$

Multiply $z$ by $z^{2}$ by adding the exponents.

Move $z^{2}$ .

$z^{2} z \sqrt{z} \sqrt[4]{z}$

Multiply $z^{2}$ by $z$ .

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

$z^{2} z^{1} \sqrt{z} \sqrt[4]{z}$

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

$z^{2 + 1} \sqrt{z} \sqrt[4]{z}$

Add $2$ and $1$ .

$z^{3} \sqrt{z} \sqrt[4]{z}$

Multiply $z^{3} \sqrt{z} \sqrt[4]{z}$ .

Rewrite the expression using the least common index of $4$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z}$ as $z^{\frac{1}{2}}$ .

$z^{3} (\sqrt[4]{z} z^{\frac{1}{2}})$

Rewrite $z^{\frac{1}{2}}$ as $z^{\frac{2}{4}}$ .

$z^{3} (\sqrt[4]{z} z^{\frac{2}{4}})$

Rewrite $z^{\frac{2}{4}}$ as $\sqrt[4]{z^{2}}$ .

$z^{3} (\sqrt[4]{z} \sqrt[4]{z^{2}})$

Combine using the product rule for radicals.

$z^{3} \sqrt[4]{z \cdot z^{2}}$

Multiply $z$ by $z^{2}$ by adding the exponents.

Multiply $z$ by $z^{2}$ .

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

$z^{3} \sqrt[4]{z^{1} z^{2}}$

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

$z^{3} \sqrt[4]{z^{1 + 2}}$

Add $1$ and $2$ .

$z^{3} \sqrt[4]{z^{3}}$

Simplify the Radical Expression ( square root of z^9)/( fourth root of z^3)

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