Simplify fourth root of 5^5* sixth root of 5^5

$\sqrt[4]{5^{5}} \cdot \sqrt[6]{5^{5}}$

Step 1

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

$\sqrt[4]{3125} \cdot \sqrt[6]{5^{5}}$

Step 2

Rewrite $3125$ as $5^{4} \cdot 5$ .

Factor $625$ out of $3125$ .

$\sqrt[4]{625 (5)} \cdot \sqrt[6]{5^{5}}$

Rewrite $625$ as $5^{4}$ .

$\sqrt[4]{5^{4} \cdot 5} \cdot \sqrt[6]{5^{5}}$

Step 3

Pull terms out from under the radical.

$5 \sqrt[4]{5} \cdot \sqrt[6]{5^{5}}$

Step 4

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

$5 \sqrt[4]{5} \cdot \sqrt[6]{3125}$

Step 5

Multiply $5 \sqrt[4]{5} \sqrt[6]{3125}$ .

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{3125}$ as $3125^{\frac{1}{6}}$ .

$5 (3125^{\frac{1}{6}} \sqrt[4]{5})$

Rewrite $3125^{\frac{1}{6}}$ as $3125^{\frac{2}{12}}$ .

$5 (3125^{\frac{2}{12}} \sqrt[4]{5})$

Rewrite $3125^{\frac{2}{12}}$ as $\sqrt[12]{3125^{2}}$ .

$5 (\sqrt[12]{3125^{2}} \sqrt[4]{5})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{5}$ as $5^{\frac{1}{4}}$ .

$5 (\sqrt[12]{3125^{2}} \cdot 5^{\frac{1}{4}})$

Rewrite $5^{\frac{1}{4}}$ as $5^{\frac{3}{12}}$ .

$5 (\sqrt[12]{3125^{2}} \cdot 5^{\frac{3}{12}})$

Rewrite $5^{\frac{3}{12}}$ as $\sqrt[12]{5^{3}}$ .

$5 (\sqrt[12]{3125^{2}} \sqrt[12]{5^{3}})$

Combine using the product rule for radicals.

$5 \sqrt[12]{3125^{2} \cdot 5^{3}}$

Rewrite $3125$ as $5^{5}$ .

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

Multiply the exponents in ${(5^{5})}^{2}$ .

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

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

Multiply $5$ by $2$ .

$5 \sqrt[12]{5^{10} \cdot 5^{3}}$

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

$5 \sqrt[12]{5^{10 + 3}}$

Add $10$ and $3$ .

$5 \sqrt[12]{5^{13}}$

Step 6

Raise $5$ to the power of $13$ .

$5 \sqrt[12]{1220703125}$

Step 7

Rewrite $1220703125$ as $5^{12} \cdot 5$ .

Factor $244140625$ out of $1220703125$ .

$5 \sqrt[12]{244140625 (5)}$

Rewrite $244140625$ as $5^{12}$ .

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

Step 8

Pull terms out from under the radical.

$5 (5 \sqrt[12]{5})$

Step 9

Multiply $5$ by $5$ .

$25 \sqrt[12]{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$25 \sqrt[12]{5}$

Decimal Form:

$28.58824590 \dots$

Simplify fourth root of 5^5* sixth root of 5^5

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