Simplify fourth root of r^11 cube root of r^5

$\sqrt[4]{r^{11}} \sqrt[3]{r^{5}}$

Step 1

Rewrite $r^{11}$ as ${(r^{2})}^{4} r^{3}$ .

Factor out $r^{8}$ .

$\sqrt[4]{r^{8} r^{3}} \sqrt[3]{r^{5}}$

Rewrite $r^{8}$ as ${(r^{2})}^{4}$ .

$\sqrt[4]{{(r^{2})}^{4} r^{3}} \sqrt[3]{r^{5}}$

Step 2

Pull terms out from under the radical.

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

Step 3

Factor out $r^{3}$ .

$r^{2} \sqrt[4]{r^{3}} \sqrt[3]{r^{3} r^{2}}$

Step 4

Pull terms out from under the radical.

$r^{2} \sqrt[4]{r^{3}} (r \sqrt[3]{r^{2}})$

Step 5

Multiply $r^{2}$ by $r$ by adding the exponents.

Move $r$ .

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

Multiply $r$ by $r^{2}$ .

Raise $r$ to the power of $1$ .

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

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

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

Add $1$ and $2$ .

$r^{3} \sqrt[4]{r^{3}} \sqrt[3]{r^{2}}$

Step 6

Multiply $r^{3} \sqrt[4]{r^{3}} \sqrt[3]{r^{2}}$ .

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

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

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

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

$r^{3} (r^{\frac{8}{12}} \sqrt[4]{r^{3}})$

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

$r^{3} (\sqrt[12]{r^{8}} \sqrt[4]{r^{3}})$

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

$r^{3} (\sqrt[12]{r^{8}} r^{\frac{3}{4}})$

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

$r^{3} (\sqrt[12]{r^{8}} r^{\frac{9}{12}})$

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

$r^{3} (\sqrt[12]{r^{8}} \sqrt[12]{r^{9}})$

Combine using the product rule for radicals.

$r^{3} \sqrt[12]{r^{8} r^{9}}$

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

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

$r^{3} \sqrt[12]{r^{8 + 9}}$

Add $8$ and $9$ .

$r^{3} \sqrt[12]{r^{17}}$

Step 7

Factor out $r^{12}$ .

$r^{3} \sqrt[12]{r^{12} r^{5}}$

Step 8

Pull terms out from under the radical.

$r^{3} (r \sqrt[12]{r^{5}})$

Step 9

Multiply $r^{3}$ by $r$ by adding the exponents.

Move $r$ .

$r \cdot r^{3} \sqrt[12]{r^{5}}$

Multiply $r$ by $r^{3}$ .

Raise $r$ to the power of $1$ .

$r^{1} r^{3} \sqrt[12]{r^{5}}$

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

$r^{1 + 3} \sqrt[12]{r^{5}}$

Add $1$ and $3$ .

$r^{4} \sqrt[12]{r^{5}}$

Simplify fourth root of r^11 cube root of r^5

Related Questions

Support LCMGCF.com by disabling your adblocker.