Simplify ( fifth root of c^7)/( cube root of c^2)

$\frac{\sqrt[5]{c^{7}}}{\sqrt[3]{c^{2}}}$

Step 1

Simplify the numerator.

Factor out $c^{5}$ .

$\frac{\sqrt[5]{c^{5} c^{2}}}{\sqrt[3]{c^{2}}}$

Pull terms out from under the radical.

$\frac{c \sqrt[5]{c^{2}}}{\sqrt[3]{c^{2}}}$

Step 2

Multiply $\frac{c \sqrt[5]{c^{2}}}{\sqrt[3]{c^{2}}}$ by $\frac{{\sqrt[3]{c^{2}}}^{2}}{{\sqrt[3]{c^{2}}}^{2}}$ .

$\frac{c \sqrt[5]{c^{2}}}{\sqrt[3]{c^{2}}} \cdot \frac{{\sqrt[3]{c^{2}}}^{2}}{{\sqrt[3]{c^{2}}}^{2}}$

Step 3

Simplify terms.

Combine and simplify the denominator.

Multiply $\frac{c \sqrt[5]{c^{2}}}{\sqrt[3]{c^{2}}}$ by $\frac{{\sqrt[3]{c^{2}}}^{2}}{{\sqrt[3]{c^{2}}}^{2}}$ .

$\frac{c \sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2}}{\sqrt[3]{c^{2}} {\sqrt[3]{c^{2}}}^{2}}$

Raise $\sqrt[3]{c^{2}}$ to the power of $1$ .

$\frac{c \sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2}}{{\sqrt[3]{c^{2}}}^{1} {\sqrt[3]{c^{2}}}^{2}}$

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

$\frac{c \sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2}}{{\sqrt[3]{c^{2}}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{c \sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2}}{{\sqrt[3]{c^{2}}}^{3}}$

Rewrite ${\sqrt[3]{c^{2}}}^{3}$ as $c^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{c^{2}}$ as $c^{\frac{2}{3}}$ .

$\frac{c \sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2}}{{(c^{\frac{2}{3}})}^{3}}$

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

$\frac{c \sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2}}{c^{\frac{2}{3} \cdot 3}}$

Combine $\frac{2}{3}$ and $3$ .

$\frac{c \sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2}}{c^{\frac{2 \cdot 3}{3}}}$

Multiply $2$ by $3$ .

$\frac{c \sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2}}{c^{\frac{6}{3}}}$

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6$ .

$\frac{c \sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2}}{c^{\frac{3 \cdot 2}{3}}}$

Cancel the common factors.

Factor $3$ out of $3$ .

$\frac{c \sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2}}{c^{\frac{3 \cdot 2}{3 (1)}}}$

Cancel the common factor.

$\frac{c \sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2}}{c^{\frac{3 \cdot 2}{3 \cdot 1}}}$

Rewrite the expression.

$\frac{c \sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2}}{c^{\frac{2}{1}}}$

Divide $2$ by $1$ .

$\frac{c \sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2}}{c^{2}}$

Cancel the common factor of $c$ and $c^{2}$ .

Factor $c$ out of $c \sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2}$ .

$\frac{c (\sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2})}{c^{2}}$

Cancel the common factors.

Factor $c$ out of $c^{2}$ .

$\frac{c (\sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2})}{c \cdot c}$

Cancel the common factor.

$\frac{c (\sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2})}{c \cdot c}$

Rewrite the expression.

$\frac{\sqrt[5]{c^{2}} {\sqrt[3]{c^{2}}}^{2}}{c}$

Step 4

Simplify the numerator.

Rewrite ${\sqrt[3]{c^{2}}}^{2}$ as ${({(c^{2})}^{2})}^{\frac{1}{3}}$ .

$\frac{\sqrt[5]{c^{2}} \sqrt[3]{{(c^{2})}^{2}}}{c}$

Multiply the exponents in ${(c^{2})}^{2}$ .

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

$\frac{\sqrt[5]{c^{2}} \sqrt[3]{c^{2 \cdot 2}}}{c}$

Multiply $2$ by $2$ .

$\frac{\sqrt[5]{c^{2}} \sqrt[3]{c^{4}}}{c}$

Factor out $c^{3}$ .

$\frac{\sqrt[5]{c^{2}} \sqrt[3]{c^{3} c}}{c}$

Pull terms out from under the radical.

$\frac{\sqrt[5]{c^{2}} c \sqrt[3]{c}}{c}$

Combine exponents.

Rewrite the expression using the least common index of $15$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{c^{2}}$ as $c^{\frac{2}{5}}$ .

$\frac{c (c^{\frac{2}{5}} \sqrt[3]{c})}{c}$

Rewrite $c^{\frac{2}{5}}$ as $c^{\frac{6}{15}}$ .

$\frac{c (c^{\frac{6}{15}} \sqrt[3]{c})}{c}$

Rewrite $c^{\frac{6}{15}}$ as $\sqrt[15]{c^{6}}$ .

$\frac{c (\sqrt[15]{c^{6}} \sqrt[3]{c})}{c}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{c}$ as $c^{\frac{1}{3}}$ .

$\frac{c (\sqrt[15]{c^{6}} c^{\frac{1}{3}})}{c}$

Rewrite $c^{\frac{1}{3}}$ as $c^{\frac{5}{15}}$ .

$\frac{c (\sqrt[15]{c^{6}} c^{\frac{5}{15}})}{c}$

Rewrite $c^{\frac{5}{15}}$ as $\sqrt[15]{c^{5}}$ .

$\frac{c (\sqrt[15]{c^{6}} \sqrt[15]{c^{5}})}{c}$

Combine using the product rule for radicals.

$\frac{c \sqrt[15]{c^{6} c^{5}}}{c}$

Multiply $c^{6}$ by $c^{5}$ by adding the exponents.

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

$\frac{c \sqrt[15]{c^{6 + 5}}}{c}$

Add $6$ and $5$ .

$\frac{c \sqrt[15]{c^{11}}}{c}$

Step 5

Cancel the common factor of $c$ .

Cancel the common factor.

$\frac{c \sqrt[15]{c^{11}}}{c}$

Divide $\sqrt[15]{c^{11}}$ by $1$ .

$\sqrt[15]{c^{11}}$

Simplify ( fifth root of c^7)/( cube root of c^2)

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