Simplify ((y^5z)^(1/10))/(y^(-1/2)z^(1/5))

$\frac{{(y^{5} z)}^{\frac{1}{10}}}{y^{- \frac{1}{2}} z^{\frac{1}{5}}}$

Step 1

Move $y^{- \frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(y^{5} z)}^{\frac{1}{10}} y^{\frac{1}{2}}}{z^{\frac{1}{5}}}$

Step 2

Simplify the numerator.

Apply the product rule to $y^{5} z$ .

$\frac{{(y^{5})}^{\frac{1}{10}} z^{\frac{1}{10}} y^{\frac{1}{2}}}{z^{\frac{1}{5}}}$

Multiply the exponents in ${(y^{5})}^{\frac{1}{10}}$ .

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

$\frac{y^{5 (\frac{1}{10})} z^{\frac{1}{10}} y^{\frac{1}{2}}}{z^{\frac{1}{5}}}$

Cancel the common factor of $5$ .

Factor $5$ out of $10$ .

$\frac{y^{5 \frac{1}{5 (2)}} z^{\frac{1}{10}} y^{\frac{1}{2}}}{z^{\frac{1}{5}}}$

Cancel the common factor.

$\frac{y^{5 \frac{1}{5 \cdot 2}} z^{\frac{1}{10}} y^{\frac{1}{2}}}{z^{\frac{1}{5}}}$

Rewrite the expression.

$\frac{y^{\frac{1}{2}} z^{\frac{1}{10}} y^{\frac{1}{2}}}{z^{\frac{1}{5}}}$

Combine exponents.

Multiply $y^{\frac{1}{2}}$ by $y^{\frac{1}{2}}$ by adding the exponents.

Move $y^{\frac{1}{2}}$ .

$\frac{y^{\frac{1}{2}} y^{\frac{1}{2}} z^{\frac{1}{10}}}{z^{\frac{1}{5}}}$

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

$\frac{y^{\frac{1}{2} + \frac{1}{2}} z^{\frac{1}{10}}}{z^{\frac{1}{5}}}$

Combine the numerators over the common denominator.

$\frac{y^{\frac{1 + 1}{2}} z^{\frac{1}{10}}}{z^{\frac{1}{5}}}$

Add $1$ and $1$ .

$\frac{y^{\frac{2}{2}} z^{\frac{1}{10}}}{z^{\frac{1}{5}}}$

Divide $2$ by $2$ .

$\frac{y^{1} z^{\frac{1}{10}}}{z^{\frac{1}{5}}}$

Simplify $y^{1} z^{\frac{1}{10}}$ .

$\frac{y z^{\frac{1}{10}}}{z^{\frac{1}{5}}}$

Step 3

Move $z^{\frac{1}{10}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{y}{z^{\frac{1}{5}} z^{- \frac{1}{10}}}$

Step 4

Multiply $z^{\frac{1}{5}}$ by $z^{- \frac{1}{10}}$ by adding the exponents.

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

$\frac{y}{z^{\frac{1}{5} - \frac{1}{10}}}$

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{y}{z^{\frac{1}{5} \cdot \frac{2}{2} - \frac{1}{10}}}$

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{5}$ by $\frac{2}{2}$ .

$\frac{y}{z^{\frac{2}{5 \cdot 2} - \frac{1}{10}}}$

Multiply $5$ by $2$ .

$\frac{y}{z^{\frac{2}{10} - \frac{1}{10}}}$

Combine the numerators over the common denominator.

$\frac{y}{z^{\frac{2 - 1}{10}}}$

Subtract $1$ from $2$ .

$\frac{y}{z^{\frac{1}{10}}}$

Simplify ((y^5z)^(1/10))/(y^(-1/2)z^(1/5))

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