Simplify the Radical Expression square root of 5^3* fifth root of 5^4

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

Step 1

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

$\sqrt{125} \cdot \sqrt[5]{5^{4}}$

Step 2

Rewrite $125$ as $5^{2} \cdot 5$ .

Factor $25$ out of $125$ .

$\sqrt{25 (5)} \cdot \sqrt[5]{5^{4}}$

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

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

Step 3

Pull terms out from under the radical.

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

Step 4

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

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

Step 5

Raise $5$ to the power of $4$ .

$5 \sqrt{5} \cdot \sqrt[5]{625}$

Step 6

Multiply $5 \sqrt{5} \sqrt[5]{625}$ .

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{625}$ as $625^{\frac{1}{5}}$ .

$5 (625^{\frac{1}{5}} \sqrt{5})$

Rewrite $625^{\frac{1}{5}}$ as $625^{\frac{2}{10}}$ .

$5 (625^{\frac{2}{10}} \sqrt{5})$

Rewrite $625^{\frac{2}{10}}$ as $\sqrt[10]{625^{2}}$ .

$5 (\sqrt[10]{625^{2}} \sqrt{5})$

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

$5 (\sqrt[10]{625^{2}} \cdot 5^{\frac{1}{2}})$

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

$5 (\sqrt[10]{625^{2}} \cdot 5^{\frac{5}{10}})$

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

$5 (\sqrt[10]{625^{2}} \sqrt[10]{5^{5}})$

Combine using the product rule for radicals.

$5 \sqrt[10]{625^{2} \cdot 5^{5}}$

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

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

Multiply the exponents in ${(5^{4})}^{2}$ .

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

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

Multiply $4$ by $2$ .

$5 \sqrt[10]{5^{8} \cdot 5^{5}}$

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

$5 \sqrt[10]{5^{8 + 5}}$

Add $8$ and $5$ .

$5 \sqrt[10]{5^{13}}$

Step 7

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

$5 \sqrt[10]{1220703125}$

Step 8

Rewrite $1220703125$ as $5^{10} \cdot 125$ .

Factor $9765625$ out of $1220703125$ .

$5 \sqrt[10]{9765625 (125)}$

Rewrite $9765625$ as $5^{10}$ .

$5 \sqrt[10]{5^{10} \cdot 125}$

Step 9

Pull terms out from under the radical.

$5 (| 5 | \sqrt[10]{125})$

Step 10

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$5 (5 \sqrt[10]{125})$

Step 11

Multiply $5$ by $5$ .

$25 \sqrt[10]{125}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$25 \sqrt[10]{125}$

Decimal Form:

$40.51641491 \dots$

Simplify the Radical Expression square root of 5^3* fifth root of 5^4

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