Basic Math Examples

$\sqrt[4]{8 a^{5} b^{3} c^{2}} \cdot \sqrt[4]{4 a b^{6} c^{3}}$

Step 1

Rewrite $8 a^{5} b^{3} c^{2}$ as $a^{4} \cdot (8 a b^{3} c^{2})$ .

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Step 1.1

Factor out $a^{4}$ .

$\sqrt[4]{8 (a^{4} a) b^{3} c^{2}} \cdot \sqrt[4]{4 a b^{6} c^{3}}$

Step 1.2

Reorder $8$ and $a^{4}$ .

$\sqrt[4]{a^{4} \cdot 8 a b^{3} c^{2}} \cdot \sqrt[4]{4 a b^{6} c^{3}}$

Step 1.3

Add parentheses.

$\sqrt[4]{a^{4} \cdot 8 a (b^{3} c^{2})} \cdot \sqrt[4]{4 a b^{6} c^{3}}$

Step 1.4

Add parentheses.

$\sqrt[4]{a^{4} \cdot 8 (a (b^{3} c^{2}))} \cdot \sqrt[4]{4 a b^{6} c^{3}}$

Step 1.5

Add parentheses.

$\sqrt[4]{a^{4} \cdot (8 a b^{3} c^{2})} \cdot \sqrt[4]{4 a b^{6} c^{3}}$

Step 2

Pull terms out from under the radical.

$a \sqrt[4]{8 a b^{3} c^{2}} \cdot \sqrt[4]{4 a b^{6} c^{3}}$

Step 3

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

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Step 3.1

Factor out $b^{4}$ .

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

Step 3.2

Move $a$ .

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

Step 3.3

Reorder $4$ and $b^{4}$ .

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

Step 3.4

Add parentheses.

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

Step 3.5

Add parentheses.

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

Step 3.6

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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Step 5.1

Combine using the product rule for radicals.

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

Step 5.2

Multiply $4$ by $8$ .

$a (b \sqrt[4]{32 a b^{3} c^{2} (a b^{2} c^{3})})$

Step 5.3

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

$a (b \sqrt[4]{32 (a^{1} a) b^{3} c^{2} (b^{2} c^{3})})$

Step 5.4

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

$a (b \sqrt[4]{32 (a^{1} a^{1}) b^{3} c^{2} (b^{2} c^{3})})$

Step 5.5

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

$a (b \sqrt[4]{32 a^{1 + 1} b^{3} c^{2} (b^{2} c^{3})})$

Step 5.6

Add $1$ and $1$ .

$a (b \sqrt[4]{32 a^{2} b^{3} c^{2} (b^{2} c^{3})})$

Step 5.7

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

$a (b \sqrt[4]{32 a^{2} b^{2 + 3} c^{2} c^{3}})$

Step 5.8

Add $2$ and $3$ .

$a (b \sqrt[4]{32 a^{2} b^{5} c^{2} c^{3}})$

Step 5.9

Multiply $c^{2}$ by $c^{3}$ by adding the exponents.

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Step 5.9.1

Move $c^{3}$ .

$a (b \sqrt[4]{32 a^{2} b^{5} (c^{3} c^{2})})$

Step 5.9.2

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

$a (b \sqrt[4]{32 a^{2} b^{5} c^{3 + 2}})$

Step 5.9.3

Add $3$ and $2$ .

$a (b \sqrt[4]{32 a^{2} b^{5} c^{5}})$

Step 6

Rewrite $32 a^{2} b^{5} c^{5}$ as ${(2 b c)}^{4} \cdot (2 a^{2} b c)$ .

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Step 6.1

Factor $16$ out of $32$ .

$a (b \sqrt[4]{16 (2) a^{2} b^{5} c^{5}})$

Step 6.2

Rewrite $16$ as $2^{4}$ .

$a (b \sqrt[4]{2^{4} \cdot 2 a^{2} b^{5} c^{5}})$

Step 6.3

Factor out $b^{4}$ .

$a (b \sqrt[4]{2^{4} \cdot 2 a^{2} (b^{4} b) c^{5}})$

Step 6.4

Factor out $c^{4}$ .

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

Step 6.5

Move $b$ .

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

Step 6.6

Move $a^{2}$ .

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

Step 6.7

Move $2$ .

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

Step 6.8

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

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

Step 6.9

Add parentheses.

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

Step 6.10

Add parentheses.

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

Step 6.11

Add parentheses.

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

Step 7

Pull terms out from under the radical.

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

Step 8

Simplify the expression.

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Step 8.1

Rewrite using the commutative property of multiplication.

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

Step 8.2

Multiply $b$ by $b$ .

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