Simplify (y^3z^-5)^(23/30)(y^4)^6

${(y^{3} z^{- 5})}^{\frac{23}{30}} {(y^{4})}^{6}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(y^{3} \frac{1}{z^{5}})}^{\frac{23}{30}} {(y^{4})}^{6}$

Step 2

Combine $y^{3}$ and $\frac{1}{z^{5}}$ .

${(\frac{y^{3}}{z^{5}})}^{\frac{23}{30}} {(y^{4})}^{6}$

Step 3

Apply basic rules of exponents.

Apply the product rule to $\frac{y^{3}}{z^{5}}$ .

$\frac{{(y^{3})}^{\frac{23}{30}}}{{(z^{5})}^{\frac{23}{30}}} {(y^{4})}^{6}$

Multiply the exponents in ${(y^{3})}^{\frac{23}{30}}$ .

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

$\frac{y^{3 (\frac{23}{30})}}{{(z^{5})}^{\frac{23}{30}}} {(y^{4})}^{6}$

Cancel the common factor of $3$ .

Factor $3$ out of $30$ .

$\frac{y^{3 \frac{23}{3 (10)}}}{{(z^{5})}^{\frac{23}{30}}} {(y^{4})}^{6}$

Cancel the common factor.

$\frac{y^{3 \frac{23}{3 \cdot 10}}}{{(z^{5})}^{\frac{23}{30}}} {(y^{4})}^{6}$

Rewrite the expression.

$\frac{y^{\frac{23}{10}}}{{(z^{5})}^{\frac{23}{30}}} {(y^{4})}^{6}$

Multiply the exponents in ${(z^{5})}^{\frac{23}{30}}$ .

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

$\frac{y^{\frac{23}{10}}}{z^{5 (\frac{23}{30})}} {(y^{4})}^{6}$

Cancel the common factor of $5$ .

Factor $5$ out of $30$ .

$\frac{y^{\frac{23}{10}}}{z^{5 \frac{23}{5 (6)}}} {(y^{4})}^{6}$

Cancel the common factor.

$\frac{y^{\frac{23}{10}}}{z^{5 \frac{23}{5 \cdot 6}}} {(y^{4})}^{6}$

Rewrite the expression.

$\frac{y^{\frac{23}{10}}}{z^{\frac{23}{6}}} {(y^{4})}^{6}$

Multiply the exponents in ${(y^{4})}^{6}$ .

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

$\frac{y^{\frac{23}{10}}}{z^{\frac{23}{6}}} y^{4 \cdot 6}$

Multiply $4$ by $6$ .

$\frac{y^{\frac{23}{10}}}{z^{\frac{23}{6}}} y^{24}$

Step 4

Multiply $\frac{y^{\frac{23}{10}}}{z^{\frac{23}{6}}} y^{24}$ .

Combine $\frac{y^{\frac{23}{10}}}{z^{\frac{23}{6}}}$ and $y^{24}$ .

$\frac{y^{\frac{23}{10}} y^{24}}{z^{\frac{23}{6}}}$

Multiply $y^{\frac{23}{10}}$ by $y^{24}$ by adding the exponents.

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

$\frac{y^{\frac{23}{10} + 24}}{z^{\frac{23}{6}}}$

To write $24$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{y^{\frac{23}{10} + 24 \cdot \frac{10}{10}}}{z^{\frac{23}{6}}}$

Combine $24$ and $\frac{10}{10}$ .

$\frac{y^{\frac{23}{10} + \frac{24 \cdot 10}{10}}}{z^{\frac{23}{6}}}$

Combine the numerators over the common denominator.

$\frac{y^{\frac{23 + 24 \cdot 10}{10}}}{z^{\frac{23}{6}}}$

Simplify the numerator.

Multiply $24$ by $10$ .

$\frac{y^{\frac{23 + 240}{10}}}{z^{\frac{23}{6}}}$

Add $23$ and $240$ .

$\frac{y^{\frac{263}{10}}}{z^{\frac{23}{6}}}$

Simplify (y^3z^-5)^(23/30)(y^4)^6

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