Simplify z^(7/10)*10 root of z^33

$z^{\frac{7}{10}} \cdot \sqrt[10]{z^{33}}$

Rewrite $z^{33}$ as ${(z^{3})}^{10} z^{3}$ .

Factor out $z^{30}$ .

$z^{\frac{7}{10}} \cdot \sqrt[10]{z^{30} z^{3}}$

Rewrite $z^{30}$ as ${(z^{3})}^{10}$ .

$z^{\frac{7}{10}} \cdot \sqrt[10]{{(z^{3})}^{10} z^{3}}$

Pull terms out from under the radical.

$z^{\frac{7}{10}} \cdot (z^{3} \sqrt[10]{z^{3}})$

Multiply $z^{\frac{7}{10}}$ by $z^{3}$ by adding the exponents.

Move $z^{3}$ .

$z^{3} z^{\frac{7}{10}} \cdot \sqrt[10]{z^{3}}$

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

$z^{3 + \frac{7}{10}} \cdot \sqrt[10]{z^{3}}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$z^{3 \cdot \frac{10}{10} + \frac{7}{10}} \cdot \sqrt[10]{z^{3}}$

Combine $3$ and $\frac{10}{10}$ .

$z^{\frac{3 \cdot 10}{10} + \frac{7}{10}} \cdot \sqrt[10]{z^{3}}$

Combine the numerators over the common denominator.

$z^{\frac{3 \cdot 10 + 7}{10}} \cdot \sqrt[10]{z^{3}}$

Simplify the numerator.

Multiply $3$ by $10$ .

$z^{\frac{30 + 7}{10}} \cdot \sqrt[10]{z^{3}}$

Add $30$ and $7$ .

$z^{\frac{37}{10}} \cdot \sqrt[10]{z^{3}}$

Multiply $z^{\frac{37}{10}} \sqrt[10]{z^{3}}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[10]{z^{3}}$ as $z^{\frac{3}{10}}$ .

$z^{\frac{37}{10}} z^{\frac{3}{10}}$

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

$z^{\frac{37}{10} + \frac{3}{10}}$

Combine the numerators over the common denominator.

$z^{\frac{37 + 3}{10}}$

Add $37$ and $3$ .

$z^{\frac{40}{10}}$

Cancel the common factor of $40$ and $10$ .

Factor $10$ out of $40$ .

$z^{\frac{10 \cdot 4}{10}}$

Cancel the common factors.

Factor $10$ out of $10$ .

$z^{\frac{10 \cdot 4}{10 (1)}}$

Cancel the common factor.

$z^{\frac{10 \cdot 4}{10 \cdot 1}}$

Rewrite the expression.

$z^{\frac{4}{1}}$

Divide $4$ by $1$ .

$z^{4}$