Simplify ( fifth root of t^2)/( sixth root of t^2)

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

Step 1

Simplify the denominator.

Rewrite $\sqrt[6]{t^{2}}$ as $\sqrt[3]{\sqrt{t^{2}}}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

Raise $\sqrt[3]{t}$ to the power of $1$ .

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

Combine $\frac{1}{3}$ and $3$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 4

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

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

Step 5

Simplify the numerator.

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

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

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

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

$\frac{t^{\frac{6}{15}} \sqrt[3]{t^{2}}}{t}$

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

$\frac{\sqrt[15]{t^{6}} \sqrt[3]{t^{2}}}{t}$

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

$\frac{\sqrt[15]{t^{6}} t^{\frac{2}{3}}}{t}$

Rewrite $t^{\frac{2}{3}}$ as $t^{\frac{10}{15}}$ .

$\frac{\sqrt[15]{t^{6}} t^{\frac{10}{15}}}{t}$

Rewrite $t^{\frac{10}{15}}$ as $\sqrt[15]{t^{10}}$ .

$\frac{\sqrt[15]{t^{6}} \sqrt[15]{t^{10}}}{t}$

Combine using the product rule for radicals.

$\frac{\sqrt[15]{t^{6} t^{10}}}{t}$

Multiply $t^{6}$ by $t^{10}$ by adding the exponents.

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

$\frac{\sqrt[15]{t^{6 + 10}}}{t}$

Add $6$ and $10$ .

$\frac{\sqrt[15]{t^{16}}}{t}$

Step 6

Simplify the numerator.

Factor out $t^{15}$ .

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

Pull terms out from under the radical.

$\frac{t \sqrt[15]{t}}{t}$

Step 7

Cancel the common factor of $t$ .

Cancel the common factor.

$\frac{t \sqrt[15]{t}}{t}$

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

$\sqrt[15]{t}$

Simplify ( fifth root of t^2)/( sixth root of t^2)

Related Questions

Support LCMGCF.com by disabling your adblocker.