Simplify cube root of 5^2* square root of 5

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

Step 1

Raise $5$ to the power of $2$ .

$\sqrt[3]{25} \cdot \sqrt{5}$

Step 2

Multiply $\sqrt[3]{25} \sqrt{5}$ .

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{25}$ as $25^{\frac{1}{3}}$ .

$25^{\frac{1}{3}} \sqrt{5}$

Rewrite $25^{\frac{1}{3}}$ as $25^{\frac{2}{6}}$ .

$25^{\frac{2}{6}} \sqrt{5}$

Rewrite $25^{\frac{2}{6}}$ as $\sqrt[6]{25^{2}}$ .

$\sqrt[6]{25^{2}} \sqrt{5}$

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

$\sqrt[6]{25^{2}} \cdot 5^{\frac{1}{2}}$

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

$\sqrt[6]{25^{2}} \cdot 5^{\frac{3}{6}}$

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

$\sqrt[6]{25^{2}} \sqrt[6]{5^{3}}$

Combine using the product rule for radicals.

$\sqrt[6]{25^{2} \cdot 5^{3}}$

Rewrite $25$ as $5^{2}$ .

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

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

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

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

Multiply $2$ by $2$ .

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

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

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

Add $4$ and $3$ .

$\sqrt[6]{5^{7}}$

Step 3

Raise $5$ to the power of $7$ .

$\sqrt[6]{78125}$

Step 4

Rewrite $78125$ as $5^{6} \cdot 5$ .

Factor $15625$ out of $78125$ .

$\sqrt[6]{15625 (5)}$

Rewrite $15625$ as $5^{6}$ .

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

Step 5

Pull terms out from under the radical.

$5 \sqrt[6]{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$5 \sqrt[6]{5}$

Decimal Form:

$6.53830243 \dots$

Simplify cube root of 5^2* square root of 5

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