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

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

Step 1

Simplify the numerator.

Factor out $t^{3}$ .

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

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Simplify terms.

Combine and simplify the denominator.

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

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

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 $t^{4}$ .

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 $5$ .

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.

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 $t^{4}$ .

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.

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.

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

Step 4

Simplify the numerator.

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

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

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

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 ${(t^{3})}^{5} t$ .

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.

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

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

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

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

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

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

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

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

$\frac{t^{3} (\sqrt[15]{t^{5}} t^{\frac{1}{5}})}{t^{3}}$

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

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

Rewrite $t^{\frac{3}{15}}$ as $\sqrt[15]{t^{3}}$ .

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

Combine using the product rule for radicals.

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

Multiply $t^{5}$ by $t^{3}$ by adding the exponents.

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

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

Add $5$ and $3$ .

$\frac{t^{3} \sqrt[15]{t^{8}}}{t^{3}}$

Step 5

Cancel the common factor of $t^{3}$ .

Cancel the common factor.

$\frac{t^{3} \sqrt[15]{t^{8}}}{t^{3}}$

Divide $\sqrt[15]{t^{8}}$ by $1$ .

$\sqrt[15]{t^{8}}$

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

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