Simplify square root of 5* cube root of 5* fourth root of 5

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

Multiply $\sqrt{5} \sqrt[3]{5}$ .

Rewrite the expression using the least common index of $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$5^{\frac{1}{2}} \sqrt[3]{5} \cdot \sqrt[4]{5}$

Rewrite $5^{\frac{1}{2}}$ as $5^{\frac{3}{6}}$ .

$5^{\frac{3}{6}} \sqrt[3]{5} \cdot \sqrt[4]{5}$

Rewrite $5^{\frac{3}{6}}$ as $\sqrt[6]{5^{3}}$ .

$\sqrt[6]{5^{3}} \sqrt[3]{5} \cdot \sqrt[4]{5}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$\sqrt[6]{5^{3}} \cdot 5^{\frac{1}{3}} \cdot \sqrt[4]{5}$

Rewrite $5^{\frac{1}{3}}$ as $5^{\frac{2}{6}}$ .

$\sqrt[6]{5^{3}} \cdot 5^{\frac{2}{6}} \cdot \sqrt[4]{5}$

Rewrite $5^{\frac{2}{6}}$ as $\sqrt[6]{5^{2}}$ .

$\sqrt[6]{5^{3}} \sqrt[6]{5^{2}} \cdot \sqrt[4]{5}$

Combine using the product rule for radicals.

$\sqrt[6]{5^{3} \cdot 5^{2}} \cdot \sqrt[4]{5}$

Multiply $5^{3}$ by $5^{2}$ by adding the exponents.

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

$\sqrt[6]{5^{3 + 2}} \cdot \sqrt[4]{5}$

Add $3$ and $2$ .

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

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

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

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

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 $3125^{\frac{1}{6}}$ .

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

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

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

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

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

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

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

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

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

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

Multiply $5$ by $2$ .

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

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

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

Add $10$ and $3$ .

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

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

$\sqrt[12]{1220703125}$

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

Factor $244140625$ out of $1220703125$ .

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

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

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

Pull terms out from under the radical.

$5 \sqrt[12]{5}$

The result can be shown in multiple forms.

Exact Form:

$5 \sqrt[12]{5}$

Decimal Form:

$5.71764918 \dots$

Simplify square root of 5* cube root of 5* fourth root of 5

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