Basic Math Examples

$\sqrt[4]{3 a^{2}} \cdot \sqrt[3]{9 a^{8}}$

Step 1

Rewrite $9 a^{8}$ as ${(a^{2})}^{3} \cdot (9 a^{2})$ .

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Step 1.1

Factor out $a^{6}$ .

$\sqrt[4]{3 a^{2}} \cdot \sqrt[3]{9 (a^{6} a^{2})}$

Step 1.2

Rewrite $a^{6}$ as ${(a^{2})}^{3}$ .

$\sqrt[4]{3 a^{2}} \cdot \sqrt[3]{9 ({(a^{2})}^{3} a^{2})}$

Step 1.3

Reorder $9$ and ${(a^{2})}^{3}$ .

$\sqrt[4]{3 a^{2}} \cdot \sqrt[3]{{(a^{2})}^{3} \cdot 9 a^{2}}$

Step 1.4

Add parentheses.

$\sqrt[4]{3 a^{2}} \cdot \sqrt[3]{{(a^{2})}^{3} \cdot (9 a^{2})}$

Step 2

Pull terms out from under the radical.

$\sqrt[4]{3 a^{2}} \cdot (a^{2} \sqrt[3]{9 a^{2}})$

Step 3

Multiply $\sqrt[4]{3 a^{2}} (a^{2} \sqrt[3]{9 a^{2}})$ .

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Step 3.1

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

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3 a^{2}}$ as ${(3 a^{2})}^{\frac{1}{4}}$ .

$a^{2} ({(3 a^{2})}^{\frac{1}{4}} \sqrt[3]{9 a^{2}})$

Step 3.1.2

Rewrite ${(3 a^{2})}^{\frac{1}{4}}$ as ${(3 a^{2})}^{\frac{3}{12}}$ .

$a^{2} ({(3 a^{2})}^{\frac{3}{12}} \sqrt[3]{9 a^{2}})$

Step 3.1.3

Rewrite ${(3 a^{2})}^{\frac{3}{12}}$ as $\sqrt[12]{{(3 a^{2})}^{3}}$ .

$a^{2} (\sqrt[12]{{(3 a^{2})}^{3}} \sqrt[3]{9 a^{2}})$

Step 3.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9 a^{2}}$ as ${(9 a^{2})}^{\frac{1}{3}}$ .

$a^{2} (\sqrt[12]{{(3 a^{2})}^{3}} {(9 a^{2})}^{\frac{1}{3}})$

Step 3.1.5

Rewrite ${(9 a^{2})}^{\frac{1}{3}}$ as ${(9 a^{2})}^{\frac{4}{12}}$ .

$a^{2} (\sqrt[12]{{(3 a^{2})}^{3}} {(9 a^{2})}^{\frac{4}{12}})$

Step 3.1.6

Rewrite ${(9 a^{2})}^{\frac{4}{12}}$ as $\sqrt[12]{{(9 a^{2})}^{4}}$ .

$a^{2} (\sqrt[12]{{(3 a^{2})}^{3}} \sqrt[12]{{(9 a^{2})}^{4}})$

Step 3.2

Combine using the product rule for radicals.

$a^{2} \sqrt[12]{{(3 a^{2})}^{3} {(9 a^{2})}^{4}}$

Step 3.3

Apply the product rule to $3 a^{2}$ .

$a^{2} \sqrt[12]{3^{3} {(a^{2})}^{3} {(9 a^{2})}^{4}}$

Step 3.4

Apply the product rule to $9 a^{2}$ .

$a^{2} \sqrt[12]{3^{3} {(a^{2})}^{3} \cdot 9^{4} {(a^{2})}^{4}}$

Step 3.5

Combine exponents.

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Step 3.5.1

Rewrite $9$ as $3^{2}$ .

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

Step 3.5.2

Multiply the exponents in ${(3^{2})}^{4}$ .

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Step 3.5.2.1

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

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

Step 3.5.2.2

Multiply $2$ by $4$ .

$a^{2} \sqrt[12]{3^{8} \cdot 3^{3} {(a^{2})}^{3} {(a^{2})}^{4}}$

Step 3.5.3

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

$a^{2} \sqrt[12]{3^{8 + 3} {(a^{2})}^{3} {(a^{2})}^{4}}$

Step 3.5.4

Add $8$ and $3$ .

$a^{2} \sqrt[12]{3^{11} {(a^{2})}^{3} {(a^{2})}^{4}}$

Step 3.5.5

Multiply ${(a^{2})}^{3}$ by ${(a^{2})}^{4}$ by adding the exponents.

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Step 3.5.5.1

Move ${(a^{2})}^{4}$ .

$a^{2} \sqrt[12]{3^{11} ({(a^{2})}^{4} {(a^{2})}^{3})}$

Step 3.5.5.2

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

$a^{2} \sqrt[12]{3^{11} {(a^{2})}^{4 + 3}}$

Step 3.5.5.3

Add $4$ and $3$ .

$a^{2} \sqrt[12]{3^{11} {(a^{2})}^{7}}$

Step 3.6

Raise $3$ to the power of $11$ .

$a^{2} \sqrt[12]{177147 {(a^{2})}^{7}}$

Step 3.7

Multiply the exponents in ${(a^{2})}^{7}$ .

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Step 3.7.1

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

$a^{2} \sqrt[12]{177147 a^{2 \cdot 7}}$

Step 3.7.2

Multiply $2$ by $7$ .

$a^{2} \sqrt[12]{177147 a^{14}}$

Step 4

Rewrite $177147 a^{14}$ as $a^{12} \cdot (177147 a^{2})$ .

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Step 4.1

Factor out $a^{12}$ .

$a^{2} \sqrt[12]{177147 (a^{12} a^{2})}$

Step 4.2

Reorder $177147$ and $a^{12}$ .

$a^{2} \sqrt[12]{a^{12} \cdot 177147 a^{2}}$

Step 4.3

Add parentheses.

$a^{2} \sqrt[12]{a^{12} \cdot (177147 a^{2})}$

Step 5

Pull terms out from under the radical.

$a^{2} (a \sqrt[12]{177147 a^{2}})$

Step 6

Multiply $a^{2}$ by $a$ by adding the exponents.

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Step 6.1

Move $a$ .

$a \cdot a^{2} \sqrt[12]{177147 a^{2}}$

Step 6.2

Multiply $a$ by $a^{2}$ .

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Step 6.2.1

Raise $a$ to the power of $1$ .

$a^{1} a^{2} \sqrt[12]{177147 a^{2}}$

Step 6.2.2

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

$a^{1 + 2} \sqrt[12]{177147 a^{2}}$

Step 6.3

Add $1$ and $2$ .

$a^{3} \sqrt[12]{177147 a^{2}}$