Basic Math Examples

$\sqrt[4]{162 a^{7} b^{7}} \cdot \sqrt[4]{3 a^{6} b^{9}}$

Step 1

Rewrite $162 a^{7} b^{7}$ as ${(3 a b)}^{4} \cdot (2 a^{3} b^{3})$ .

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Step 1.1

Factor $81$ out of $162$ .

$\sqrt[4]{81 (2) a^{7} b^{7}} \cdot \sqrt[4]{3 a^{6} b^{9}}$

Step 1.2

Rewrite $81$ as $3^{4}$ .

$\sqrt[4]{3^{4} \cdot 2 a^{7} b^{7}} \cdot \sqrt[4]{3 a^{6} b^{9}}$

Step 1.3

Factor out $a^{4}$ .

$\sqrt[4]{3^{4} \cdot 2 (a^{4} a^{3}) b^{7}} \cdot \sqrt[4]{3 a^{6} b^{9}}$

Step 1.4

Factor out $b^{4}$ .

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

Step 1.5

Move $a^{3}$ .

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

Step 1.6

Move $2$ .

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

Step 1.7

Rewrite $(3^{4} (a^{4})) b^{4}$ as ${(3 a b)}^{4}$ .

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

Step 1.8

Add parentheses.

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

Step 1.9

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

Rewrite $3 a^{6} b^{9}$ as ${(a b^{2})}^{4} \cdot (3 a^{2} b)$ .

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Step 3.1

Factor out $a^{4}$ .

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

Step 3.2

Factor out $b^{8}$ .

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

Step 3.3

Rewrite $b^{8}$ as ${(b^{2})}^{4}$ .

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

Step 3.4

Move $a^{2}$ .

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

Step 3.5

Move $3$ .

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

Step 3.6

Rewrite $(a^{4}) {(b^{2})}^{4}$ as ${(a b^{2})}^{4}$ .

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

Step 3.7

Add parentheses.

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

Step 3.8

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

Multiply $a$ by $a$ by adding the exponents.

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Step 5.1

Move $a$ .

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

Step 5.2

Multiply $a$ by $a$ .

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

Step 6

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

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Step 6.1

Move $b^{2}$ .

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

Step 6.2

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

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Step 6.2.1

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

$3 a^{2} (b^{2} b^{1}) \sqrt[4]{2 a^{3} b^{3}} \cdot \sqrt[4]{3 a^{2} b}$

Step 6.2.2

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

$3 a^{2} b^{2 + 1} \sqrt[4]{2 a^{3} b^{3}} \cdot \sqrt[4]{3 a^{2} b}$

Step 6.3

Add $2$ and $1$ .

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

Step 7

Multiply $3 a^{2} b^{3} \sqrt[4]{2 a^{3} b^{3}} \sqrt[4]{3 a^{2} b}$ .

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Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

Multiply $2$ by $3$ .

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

Step 7.3

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

$3 a^{2} b^{3} \sqrt[4]{6 a^{3 + 2} b \cdot b^{3}}$

Step 7.4

Add $3$ and $2$ .

$3 a^{2} b^{3} \sqrt[4]{6 a^{5} b \cdot b^{3}}$

Step 7.5

Multiply $b$ by $b^{3}$ by adding the exponents.

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Step 7.5.1

Move $b^{3}$ .

$3 a^{2} b^{3} \sqrt[4]{6 a^{5} (b^{3} b)}$

Step 7.5.2

Multiply $b^{3}$ by $b$ .

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Step 7.5.2.1

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

$3 a^{2} b^{3} \sqrt[4]{6 a^{5} (b^{3} b^{1})}$

Step 7.5.2.2

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

$3 a^{2} b^{3} \sqrt[4]{6 a^{5} b^{3 + 1}}$

Step 7.5.3

Add $3$ and $1$ .

$3 a^{2} b^{3} \sqrt[4]{6 a^{5} b^{4}}$

Step 8

Rewrite $6 a^{5} b^{4}$ as ${(a b)}^{4} \cdot (6 a)$ .

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Step 8.1

Factor out $a^{4}$ .

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

Step 8.2

Move $a$ .

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

Step 8.3

Move $6$ .

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

Step 8.4

Rewrite $(a^{4}) b^{4}$ as ${(a b)}^{4}$ .

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

Step 8.5

Add parentheses.

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

Step 9

Pull terms out from under the radical.

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

Step 10

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

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Step 10.1

Move $a$ .

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

Step 10.2

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

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Step 10.2.1

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

$3 (a^{1} a^{2}) b^{3} (b \sqrt[4]{6 a})$

Step 10.2.2

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

$3 a^{1 + 2} b^{3} (b \sqrt[4]{6 a})$

Step 10.3

Add $1$ and $2$ .

$3 a^{3} b^{3} (b \sqrt[4]{6 a})$

Step 11

Multiply $b^{3}$ by $b$ by adding the exponents.

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Step 11.1

Move $b$ .

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

Step 11.2

Multiply $b$ by $b^{3}$ .

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Step 11.2.1

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

$3 a^{3} (b^{1} b^{3}) \sqrt[4]{6 a}$

Step 11.2.2

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

$3 a^{3} b^{1 + 3} \sqrt[4]{6 a}$

Step 11.3

Add $1$ and $3$ .

$3 a^{3} b^{4} \sqrt[4]{6 a}$