Simplify ( cube root of b^2)/( fourth root of b)

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

Step 1

Multiply $\frac{\sqrt[3]{b^{2}}}{\sqrt[4]{b}}$ by $\frac{{\sqrt[4]{b}}^{3}}{{\sqrt[4]{b}}^{3}}$ .

$\frac{\sqrt[3]{b^{2}}}{\sqrt[4]{b}} \cdot \frac{{\sqrt[4]{b}}^{3}}{{\sqrt[4]{b}}^{3}}$

Step 2

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{b^{2}}}{\sqrt[4]{b}}$ by $\frac{{\sqrt[4]{b}}^{3}}{{\sqrt[4]{b}}^{3}}$ .

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

Raise $\sqrt[4]{b}$ to the power of $1$ .

$\frac{\sqrt[3]{b^{2}} {\sqrt[4]{b}}^{3}}{{\sqrt[4]{b}}^{1} {\sqrt[4]{b}}^{3}}$

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

$\frac{\sqrt[3]{b^{2}} {\sqrt[4]{b}}^{3}}{{\sqrt[4]{b}}^{1 + 3}}$

Add $1$ and $3$ .

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{b}$ as $b^{\frac{1}{4}}$ .

$\frac{\sqrt[3]{b^{2}} {\sqrt[4]{b}}^{3}}{{(b^{\frac{1}{4}})}^{4}}$

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

$\frac{\sqrt[3]{b^{2}} {\sqrt[4]{b}}^{3}}{b^{\frac{1}{4} \cdot 4}}$

Combine $\frac{1}{4}$ and $4$ .

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{\sqrt[3]{b^{2}} {\sqrt[4]{b}}^{3}}{b^{1}}$

Simplify.

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

Step 3

Rewrite ${\sqrt[4]{b}}^{3}$ as $\sqrt[4]{b^{3}}$ .

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

Step 4

Simplify the numerator.

Rewrite the expression using the least common index of $12$ .

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

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

Rewrite $b^{\frac{2}{3}}$ as $b^{\frac{8}{12}}$ .

$\frac{b^{\frac{8}{12}} \sqrt[4]{b^{3}}}{b}$

Rewrite $b^{\frac{8}{12}}$ as $\sqrt[12]{b^{8}}$ .

$\frac{\sqrt[12]{b^{8}} \sqrt[4]{b^{3}}}{b}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{b^{3}}$ as $b^{\frac{3}{4}}$ .

$\frac{\sqrt[12]{b^{8}} b^{\frac{3}{4}}}{b}$

Rewrite $b^{\frac{3}{4}}$ as $b^{\frac{9}{12}}$ .

$\frac{\sqrt[12]{b^{8}} b^{\frac{9}{12}}}{b}$

Rewrite $b^{\frac{9}{12}}$ as $\sqrt[12]{b^{9}}$ .

$\frac{\sqrt[12]{b^{8}} \sqrt[12]{b^{9}}}{b}$

Combine using the product rule for radicals.

$\frac{\sqrt[12]{b^{8} b^{9}}}{b}$

Multiply $b^{8}$ by $b^{9}$ by adding the exponents.

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

$\frac{\sqrt[12]{b^{8 + 9}}}{b}$

Add $8$ and $9$ .

$\frac{\sqrt[12]{b^{17}}}{b}$

Step 5

Simplify the numerator.

Factor out $b^{12}$ .

$\frac{\sqrt[12]{b^{12} b^{5}}}{b}$

Pull terms out from under the radical.

$\frac{b \sqrt[12]{b^{5}}}{b}$

Step 6

Cancel the common factor of $b$ .

Cancel the common factor.

$\frac{b \sqrt[12]{b^{5}}}{b}$

Divide $\sqrt[12]{b^{5}}$ by $1$ .

$\sqrt[12]{b^{5}}$

Simplify ( cube root of b^2)/( fourth root of b)

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