Simplify fifth root of 10^2* cube root of 10^4

$\sqrt[5]{10^{2}} \cdot \sqrt[3]{10^{4}}$

Step 1

Raise $10$ to the power of $2$ .

$\sqrt[5]{100} \cdot \sqrt[3]{10^{4}}$

Step 2

Raise $10$ to the power of $4$ .

$\sqrt[5]{100} \cdot \sqrt[3]{10000}$

Step 3

Rewrite $10000$ as $10^{3} \cdot 10$ .

Factor $1000$ out of $10000$ .

$\sqrt[5]{100} \cdot \sqrt[3]{1000 (10)}$

Rewrite $1000$ as $10^{3}$ .

$\sqrt[5]{100} \cdot \sqrt[3]{10^{3} \cdot 10}$

Step 4

Pull terms out from under the radical.

$\sqrt[5]{100} \cdot (10 \sqrt[3]{10})$

Step 5

Multiply $\sqrt[5]{100} (10 \sqrt[3]{10})$ .

Rewrite the expression using the least common index of $15$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{100}$ as $100^{\frac{1}{5}}$ .

$10 (100^{\frac{1}{5}} \sqrt[3]{10})$

Rewrite $100^{\frac{1}{5}}$ as $100^{\frac{3}{15}}$ .

$10 (100^{\frac{3}{15}} \sqrt[3]{10})$

Rewrite $100^{\frac{3}{15}}$ as $\sqrt[15]{100^{3}}$ .

$10 (\sqrt[15]{100^{3}} \sqrt[3]{10})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{10}$ as $10^{\frac{1}{3}}$ .

$10 (\sqrt[15]{100^{3}} \cdot 10^{\frac{1}{3}})$

Rewrite $10^{\frac{1}{3}}$ as $10^{\frac{5}{15}}$ .

$10 (\sqrt[15]{100^{3}} \cdot 10^{\frac{5}{15}})$

Rewrite $10^{\frac{5}{15}}$ as $\sqrt[15]{10^{5}}$ .

$10 (\sqrt[15]{100^{3}} \sqrt[15]{10^{5}})$

Combine using the product rule for radicals.

$10 \sqrt[15]{100^{3} \cdot 10^{5}}$

Rewrite $100$ as $10^{2}$ .

$10 \sqrt[15]{{(10^{2})}^{3} \cdot 10^{5}}$

Multiply the exponents in ${(10^{2})}^{3}$ .

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

$10 \sqrt[15]{10^{2 \cdot 3} \cdot 10^{5}}$

Multiply $2$ by $3$ .

$10 \sqrt[15]{10^{6} \cdot 10^{5}}$

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

$10 \sqrt[15]{10^{6 + 5}}$

Add $6$ and $5$ .

$10 \sqrt[15]{10^{11}}$

Step 6

Raise $10$ to the power of $11$ .

$10 \sqrt[15]{100000000000}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$10 \sqrt[15]{100000000000}$

Decimal Form:

$54.11695265 \dots$

Simplify fifth root of 10^2* cube root of 10^4

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