Simplify the Radical Expression (p^21q^-15r^-3)^(1/3)

${(p^{21} q^{- 15} r^{- 3})}^{\frac{1}{3}}$

Step 1

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

${(p^{21} \frac{1}{q^{15}} r^{- 3})}^{\frac{1}{3}}$

Step 2

Combine $p^{21}$ and $\frac{1}{q^{15}}$ .

${(\frac{p^{21}}{q^{15}} r^{- 3})}^{\frac{1}{3}}$

Step 3

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

${(\frac{p^{21}}{q^{15}} \cdot \frac{1}{r^{3}})}^{\frac{1}{3}}$

Step 4

Combine.

${(\frac{p^{21} \cdot 1}{q^{15} r^{3}})}^{\frac{1}{3}}$

Step 5

Multiply $p^{21}$ by $1$ .

${(\frac{p^{21}}{q^{15} r^{3}})}^{\frac{1}{3}}$

Step 6

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

Apply the product rule to $\frac{p^{21}}{q^{15} r^{3}}$ .

$\frac{{(p^{21})}^{\frac{1}{3}}}{{(q^{15} r^{3})}^{\frac{1}{3}}}$

Apply the product rule to $q^{15} r^{3}$ .

$\frac{{(p^{21})}^{\frac{1}{3}}}{{(q^{15})}^{\frac{1}{3}} {(r^{3})}^{\frac{1}{3}}}$

Step 7

Multiply the exponents in ${(p^{21})}^{\frac{1}{3}}$ .

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

$\frac{p^{21 (\frac{1}{3})}}{{(q^{15})}^{\frac{1}{3}} {(r^{3})}^{\frac{1}{3}}}$

Cancel the common factor of $3$ .

Factor $3$ out of $21$ .

$\frac{p^{3 (7) \frac{1}{3}}}{{(q^{15})}^{\frac{1}{3}} {(r^{3})}^{\frac{1}{3}}}$

Cancel the common factor.

$\frac{p^{3 \cdot 7 \frac{1}{3}}}{{(q^{15})}^{\frac{1}{3}} {(r^{3})}^{\frac{1}{3}}}$

Rewrite the expression.

$\frac{p^{7}}{{(q^{15})}^{\frac{1}{3}} {(r^{3})}^{\frac{1}{3}}}$

Step 8

Simplify the denominator.

Multiply the exponents in ${(q^{15})}^{\frac{1}{3}}$ .

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

$\frac{p^{7}}{q^{15 (\frac{1}{3})} {(r^{3})}^{\frac{1}{3}}}$

Cancel the common factor of $3$ .

Factor $3$ out of $15$ .

$\frac{p^{7}}{q^{3 (5) \frac{1}{3}} {(r^{3})}^{\frac{1}{3}}}$

Cancel the common factor.

$\frac{p^{7}}{q^{3 \cdot 5 \frac{1}{3}} {(r^{3})}^{\frac{1}{3}}}$

Rewrite the expression.

$\frac{p^{7}}{q^{5} {(r^{3})}^{\frac{1}{3}}}$

Multiply the exponents in ${(r^{3})}^{\frac{1}{3}}$ .

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

$\frac{p^{7}}{q^{5} r^{3 (\frac{1}{3})}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{p^{7}}{q^{5} r^{3 (\frac{1}{3})}}$

Rewrite the expression.

$\frac{p^{7}}{q^{5} r^{1}}$

Simplify.

$\frac{p^{7}}{q^{5} r}$

Simplify the Radical Expression (p^21q^-15r^-3)^(1/3)

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