Simplify (x^15y^10z^-50)^(1/5)

${(x^{15} y^{10} z^{- 50})}^{\frac{1}{5}}$

Step 1

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

${(x^{15} y^{10} \frac{1}{z^{50}})}^{\frac{1}{5}}$

Step 2

Multiply $x^{15} y^{10} \frac{1}{z^{50}}$ .

Combine $x^{15}$ and $\frac{1}{z^{50}}$ .

${(y^{10} \frac{x^{15}}{z^{50}})}^{\frac{1}{5}}$

Combine $y^{10}$ and $\frac{x^{15}}{z^{50}}$ .

${(\frac{y^{10} x^{15}}{z^{50}})}^{\frac{1}{5}}$

Step 3

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

Apply the product rule to $\frac{y^{10} x^{15}}{z^{50}}$ .

$\frac{{(y^{10} x^{15})}^{\frac{1}{5}}}{{(z^{50})}^{\frac{1}{5}}}$

Apply the product rule to $y^{10} x^{15}$ .

$\frac{{(y^{10})}^{\frac{1}{5}} {(x^{15})}^{\frac{1}{5}}}{{(z^{50})}^{\frac{1}{5}}}$

Step 4

Simplify the numerator.

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

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

$\frac{y^{10 (\frac{1}{5})} {(x^{15})}^{\frac{1}{5}}}{{(z^{50})}^{\frac{1}{5}}}$

Cancel the common factor of $5$ .

Factor $5$ out of $10$ .

$\frac{y^{5 (2) \frac{1}{5}} {(x^{15})}^{\frac{1}{5}}}{{(z^{50})}^{\frac{1}{5}}}$

Cancel the common factor.

$\frac{y^{5 \cdot 2 \frac{1}{5}} {(x^{15})}^{\frac{1}{5}}}{{(z^{50})}^{\frac{1}{5}}}$

Rewrite the expression.

$\frac{y^{2} {(x^{15})}^{\frac{1}{5}}}{{(z^{50})}^{\frac{1}{5}}}$

Multiply the exponents in ${(x^{15})}^{\frac{1}{5}}$ .

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

$\frac{y^{2} x^{15 (\frac{1}{5})}}{{(z^{50})}^{\frac{1}{5}}}$

Cancel the common factor of $5$ .

Factor $5$ out of $15$ .

$\frac{y^{2} x^{5 (3) \frac{1}{5}}}{{(z^{50})}^{\frac{1}{5}}}$

Cancel the common factor.

$\frac{y^{2} x^{5 \cdot 3 \frac{1}{5}}}{{(z^{50})}^{\frac{1}{5}}}$

Rewrite the expression.

$\frac{y^{2} x^{3}}{{(z^{50})}^{\frac{1}{5}}}$

Step 5

Multiply the exponents in ${(z^{50})}^{\frac{1}{5}}$ .

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

$\frac{y^{2} x^{3}}{z^{50 (\frac{1}{5})}}$

Cancel the common factor of $5$ .

Factor $5$ out of $50$ .

$\frac{y^{2} x^{3}}{z^{5 (10) \frac{1}{5}}}$

Cancel the common factor.

$\frac{y^{2} x^{3}}{z^{5 \cdot 10 \frac{1}{5}}}$

Rewrite the expression.

$\frac{y^{2} x^{3}}{z^{10}}$

Simplify (x^15y^10z^-50)^(1/5)

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