Simplify ( cube root of t^4)/( fifth root of t^4)

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

Simplify the numerator.

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Factor out $t^{3}$ .

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

Pull terms out from under the radical.

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

Multiply $\frac{t \sqrt[3]{t}}{\sqrt[5]{t^{4}}}$ by $\frac{{\sqrt[5]{t^{4}}}^{4}}{{\sqrt[5]{t^{4}}}^{4}}$ .

$\frac{t \sqrt[3]{t}}{\sqrt[5]{t^{4}}} \cdot \frac{{\sqrt[5]{t^{4}}}^{4}}{{\sqrt[5]{t^{4}}}^{4}}$

Simplify terms.

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Combine and simplify the denominator.

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Multiply $\frac{t \sqrt[3]{t}}{\sqrt[5]{t^{4}}}$ and $\frac{{\sqrt[5]{t^{4}}}^{4}}{{\sqrt[5]{t^{4}}}^{4}}$ .

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

Raise $\sqrt[5]{t^{4}}$ to the power of $1$ .

$\frac{t \sqrt[3]{t} {\sqrt[5]{t^{4}}}^{4}}{{\sqrt[5]{t^{4}}}^{1} {\sqrt[5]{t^{4}}}^{4}}$

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

$\frac{t \sqrt[3]{t} {\sqrt[5]{t^{4}}}^{4}}{{\sqrt[5]{t^{4}}}^{1 + 4}}$

Add $1$ and $4$ .

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

Rewrite ${\sqrt[5]{t^{4}}}^{5}$ as $t^{4}$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{t^{4}}$ as $t^{\frac{4}{5}}$ .

$\frac{t \sqrt[3]{t} {\sqrt[5]{t^{4}}}^{4}}{{(t^{\frac{4}{5}})}^{5}}$

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

$\frac{t \sqrt[3]{t} {\sqrt[5]{t^{4}}}^{4}}{t^{\frac{4}{5} \cdot 5}}$

Combine $\frac{4}{5}$ and $5$ .

$\frac{t \sqrt[3]{t} {\sqrt[5]{t^{4}}}^{4}}{t^{\frac{4 \cdot 5}{5}}}$

Multiply $4$ by $5$ .

$\frac{t \sqrt[3]{t} {\sqrt[5]{t^{4}}}^{4}}{t^{\frac{20}{5}}}$

Cancel the common factor of $20$ and $5$ .

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Factor $5$ out of $20$ .

$\frac{t \sqrt[3]{t} {\sqrt[5]{t^{4}}}^{4}}{t^{\frac{5 \cdot 4}{5}}}$

Cancel the common factors.

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Factor $5$ out of $5$ .

$\frac{t \sqrt[3]{t} {\sqrt[5]{t^{4}}}^{4}}{t^{\frac{5 \cdot 4}{5 (1)}}}$

Cancel the common factor.

$\frac{t \sqrt[3]{t} {\sqrt[5]{t^{4}}}^{4}}{t^{\frac{5 \cdot 4}{5 \cdot 1}}}$

Rewrite the expression.

$\frac{t \sqrt[3]{t} {\sqrt[5]{t^{4}}}^{4}}{t^{\frac{4}{1}}}$

Divide $4$ by $1$ .

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

Cancel the common factor of $t$ and $t^{4}$ .

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Factor $t$ out of $t \sqrt[3]{t} {\sqrt[5]{t^{4}}}^{4}$ .

$\frac{t (\sqrt[3]{t} {\sqrt[5]{t^{4}}}^{4})}{t^{4}}$

Cancel the common factors.

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Factor $t$ out of $t^{4}$ .

$\frac{t (\sqrt[3]{t} {\sqrt[5]{t^{4}}}^{4})}{t \cdot t^{3}}$

Cancel the common factor.

$\frac{t (\sqrt[3]{t} {\sqrt[5]{t^{4}}}^{4})}{t \cdot t^{3}}$

Rewrite the expression.

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

Simplify the numerator.

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Rewrite ${\sqrt[5]{t^{4}}}^{4}$ as ${({(t^{4})}^{4})}^{\frac{1}{5}}$ .

$\frac{\sqrt[3]{t} \sqrt[5]{{(t^{4})}^{4}}}{t^{3}}$

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

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

$\frac{\sqrt[3]{t} \sqrt[5]{t^{4 \cdot 4}}}{t^{3}}$

Multiply $4$ by $4$ .

$\frac{\sqrt[3]{t} \sqrt[5]{t^{16}}}{t^{3}}$

Rewrite $t^{16}$ as ${(t^{3})}^{5} t$ .

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Factor out $t^{15}$ .

$\frac{\sqrt[3]{t} \sqrt[5]{t^{15} t}}{t^{3}}$

Rewrite $t^{15}$ as ${(t^{3})}^{5}$ .

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

Pull terms out from under the radical.

$\frac{\sqrt[3]{t} t^{3} \sqrt[5]{t}}{t^{3}}$

Combine exponents.

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

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

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

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

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{t^{\frac{1}{3} + 3 \cdot \frac{3}{3}} \sqrt[5]{t}}{t^{3}}$

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

$\frac{t^{\frac{1}{3} + \frac{3 \cdot 3}{3}} \sqrt[5]{t}}{t^{3}}$

Combine the numerators over the common denominator.

$\frac{t^{\frac{1 + 3 \cdot 3}{3}} \sqrt[5]{t}}{t^{3}}$

Simplify the numerator.

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Multiply $3$ by $3$ .

$\frac{t^{\frac{1 + 9}{3}} \sqrt[5]{t}}{t^{3}}$

Add $1$ and $9$ .

$\frac{t^{\frac{10}{3}} \sqrt[5]{t}}{t^{3}}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{t}$ as $t^{\frac{1}{5}}$ .

$\frac{t^{\frac{10}{3}} t^{\frac{1}{5}}}{t^{3}}$

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

$\frac{t^{\frac{10}{3} + \frac{1}{5}}}{t^{3}}$

To write $\frac{10}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{t^{\frac{10}{3} \cdot \frac{5}{5} + \frac{1}{5}}}{t^{3}}$

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{t^{\frac{10}{3} \cdot \frac{5}{5} + \frac{1}{5} \cdot \frac{3}{3}}}{t^{3}}$

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{10}{3}$ and $\frac{5}{5}$ .

$\frac{t^{\frac{10 \cdot 5}{3 \cdot 5} + \frac{1}{5} \cdot \frac{3}{3}}}{t^{3}}$

Multiply $3$ by $5$ .

$\frac{t^{\frac{10 \cdot 5}{15} + \frac{1}{5} \cdot \frac{3}{3}}}{t^{3}}$

Multiply $\frac{1}{5}$ and $\frac{3}{3}$ .

$\frac{t^{\frac{10 \cdot 5}{15} + \frac{3}{5 \cdot 3}}}{t^{3}}$

Multiply $5$ by $3$ .

$\frac{t^{\frac{10 \cdot 5}{15} + \frac{3}{15}}}{t^{3}}$

Combine the numerators over the common denominator.

$\frac{t^{\frac{10 \cdot 5 + 3}{15}}}{t^{3}}$

Simplify the numerator.

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Multiply $10$ by $5$ .

$\frac{t^{\frac{50 + 3}{15}}}{t^{3}}$

Add $50$ and $3$ .

$\frac{t^{\frac{53}{15}}}{t^{3}}$

Move $t^{3}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$t^{\frac{53}{15}} t^{- 3}$

Multiply $t^{\frac{53}{15}}$ by $t^{- 3}$ by adding the exponents.

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

$t^{\frac{53}{15} - 3}$

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$t^{\frac{53}{15} - 3 \cdot \frac{15}{15}}$

Combine $- 3$ and $\frac{15}{15}$ .

$t^{\frac{53}{15} + \frac{- 3 \cdot 15}{15}}$

Combine the numerators over the common denominator.

$t^{\frac{53 - 3 \cdot 15}{15}}$

Simplify the numerator.

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Multiply $- 3$ by $15$ .

$t^{\frac{53 - 45}{15}}$

Subtract $45$ from $53$ .

$t^{\frac{8}{15}}$

Simplify ( cube root of t^4)/( fifth root of t^4)