Algebra Examples

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

Step 1

Rewrite $12$ as $2^{2} \cdot 3$ .

Tap for more steps...

Step 1.1

Factor $4$ out of $12$ .

$\sqrt{4 (3)} \cdot \sqrt[5]{12^{3}}$

Step 1.2

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} \cdot 3} \cdot \sqrt[5]{12^{3}}$

Step 2

Pull terms out from under the radical.

$2 \sqrt{3} \cdot \sqrt[5]{12^{3}}$

Step 3

Raise $12$ to the power of $3$ .

$2 \sqrt{3} \cdot \sqrt[5]{1728}$

Step 4

Rewrite $1728$ as $2^{5} \cdot 54$ .

Tap for more steps...

Step 4.1

Factor $32$ out of $1728$ .

$2 \sqrt{3} \cdot \sqrt[5]{32 (54)}$

Step 4.2

Rewrite $32$ as $2^{5}$ .

$2 \sqrt{3} \cdot \sqrt[5]{2^{5} \cdot 54}$

Step 5

Pull terms out from under the radical.

$2 \sqrt{3} \cdot (2 \sqrt[5]{54})$

Step 6

Multiply $2 \sqrt{3} (2 \sqrt[5]{54})$ .

Tap for more steps...

Step 6.1

Multiply $2$ by $2$ .

$4 \sqrt{3} \sqrt[5]{54}$

Step 6.2

Rewrite the expression using the least common index of $10$ .

Tap for more steps...

Step 6.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{54}$ as $54^{\frac{1}{5}}$ .

$4 (54^{\frac{1}{5}} \sqrt{3})$

Step 6.2.2

Rewrite $54^{\frac{1}{5}}$ as $54^{\frac{2}{10}}$ .

$4 (54^{\frac{2}{10}} \sqrt{3})$

Step 6.2.3

Rewrite $54^{\frac{2}{10}}$ as $\sqrt[10]{54^{2}}$ .

$4 (\sqrt[10]{54^{2}} \sqrt{3})$

Step 6.2.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$4 (\sqrt[10]{54^{2}} \cdot 3^{\frac{1}{2}})$

Step 6.2.5

Rewrite $3^{\frac{1}{2}}$ as $3^{\frac{5}{10}}$ .

$4 (\sqrt[10]{54^{2}} \cdot 3^{\frac{5}{10}})$

Step 6.2.6

Rewrite $3^{\frac{5}{10}}$ as $\sqrt[10]{3^{5}}$ .

$4 (\sqrt[10]{54^{2}} \sqrt[10]{3^{5}})$

Step 6.3

Combine using the product rule for radicals.

$4 \sqrt[10]{54^{2} \cdot 3^{5}}$

Step 6.4

Raise $54$ to the power of $2$ .

$4 \sqrt[10]{2916 \cdot 3^{5}}$

Step 6.5

Raise $3$ to the power of $5$ .

$4 \sqrt[10]{2916 \cdot 243}$

Step 6.6

Multiply $2916$ by $243$ .

$4 \sqrt[10]{708588}$

Step 7

Rewrite $708588$ as $3^{10} \cdot 12$ .

Tap for more steps...

Step 7.1

Factor $59049$ out of $708588$ .

$4 \sqrt[10]{59049 (12)}$

Step 7.2

Rewrite $59049$ as $3^{10}$ .

$4 \sqrt[10]{3^{10} \cdot 12}$

Step 8

Pull terms out from under the radical.

$4 (3 \sqrt[10]{12})$

Step 9

Multiply $3$ by $4$ .

$12 \sqrt[10]{12}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$12 \sqrt[10]{12}$

Decimal Form:

$15.38506624 \dots$