Evaluate cube root of 5^5* square root of 5

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

Step 1

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

$\sqrt[3]{3125} \cdot \sqrt{5}$

Step 2

Rewrite $3125$ as $5^{3} \cdot 25$ .

Factor $125$ out of $3125$ .

$\sqrt[3]{125 (25)} \cdot \sqrt{5}$

Rewrite $125$ as $5^{3}$ .

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

Step 3

Pull terms out from under the radical.

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

Step 4

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

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

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

$5 (5^{\frac{1}{2}} \sqrt[3]{25})$

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

$5 (5^{\frac{3}{6}} \sqrt[3]{25})$

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

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

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

$5 (\sqrt[6]{5^{3}} \cdot 25^{\frac{1}{3}})$

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

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

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

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

Combine using the product rule for radicals.

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

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

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

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

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

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

Multiply $2$ by $2$ .

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

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

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

Add $3$ and $4$ .

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

Step 5

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

$5 \sqrt[6]{78125}$

Step 6

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

Factor $15625$ out of $78125$ .

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

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

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

Step 7

Pull terms out from under the radical.

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

Step 8

Multiply $5$ by $5$ .

$25 \sqrt[6]{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$25 \sqrt[6]{5}$

Decimal Form:

$32.69151215 \dots$

Evaluate cube root of 5^5* square root of 5

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