Evaluate cube root of 10^5* fourth root of 10^5

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

Raise $10$ to the power of $5$ .

$\sqrt[3]{100000} \cdot \sqrt[4]{10^{5}}$

Rewrite $100000$ as $10^{3} \cdot 100$ .

Factor $1000$ out of $100000$ .

$\sqrt[3]{1000 (100)} \cdot \sqrt[4]{10^{5}}$

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

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

Pull terms out from under the radical.

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

Raise $10$ to the power of $5$ .

$10 \sqrt[3]{100} \cdot \sqrt[4]{100000}$

Rewrite $100000$ as $10^{4} \cdot 10$ .

Factor $10000$ out of $100000$ .

$10 \sqrt[3]{100} \cdot \sqrt[4]{10000 (10)}$

Rewrite $10000$ as $10^{4}$ .

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

Pull terms out from under the radical.

$10 \sqrt[3]{100} \cdot (10 \sqrt[4]{10})$

Multiply $10 \sqrt[3]{100} (10 \sqrt[4]{10})$ .

Multiply $10$ by $10$ .

$100 \sqrt[3]{100} \sqrt[4]{10}$

Rewrite the expression using the least common index of $12$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{10}$ as $10^{\frac{1}{4}}$ .

$100 (10^{\frac{1}{4}} \sqrt[3]{100})$

Rewrite $10^{\frac{1}{4}}$ as $10^{\frac{3}{12}}$ .

$100 (10^{\frac{3}{12}} \sqrt[3]{100})$

Rewrite $10^{\frac{3}{12}}$ as $\sqrt[12]{10^{3}}$ .

$100 (\sqrt[12]{10^{3}} \sqrt[3]{100})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{100}$ as $100^{\frac{1}{3}}$ .

$100 (\sqrt[12]{10^{3}} \cdot 100^{\frac{1}{3}})$

Rewrite $100^{\frac{1}{3}}$ as $100^{\frac{4}{12}}$ .

$100 (\sqrt[12]{10^{3}} \cdot 100^{\frac{4}{12}})$

Rewrite $100^{\frac{4}{12}}$ as $\sqrt[12]{100^{4}}$ .

$100 (\sqrt[12]{10^{3}} \sqrt[12]{100^{4}})$

Combine using the product rule for radicals.

$100 \sqrt[12]{10^{3} \cdot 100^{4}}$

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

$100 \sqrt[12]{10^{3} \cdot {(10^{2})}^{4}}$

Multiply the exponents in ${(10^{2})}^{4}$ .

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

$100 \sqrt[12]{10^{3} \cdot 10^{2 \cdot 4}}$

Multiply $2$ by $4$ .

$100 \sqrt[12]{10^{3} \cdot 10^{8}}$

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

$100 \sqrt[12]{10^{3 + 8}}$

Add $3$ and $8$ .

$100 \sqrt[12]{10^{11}}$

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

$100 \sqrt[12]{100000000000}$

The result can be shown in multiple forms.

Exact Form:

$100 \sqrt[12]{100000000000}$

Decimal Form:

$825.40418526 \dots$

Evaluate cube root of 10^5* fourth root of 10^5

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