Algebra Examples

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

Step 1

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

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Step 1.1

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

$\sqrt{6 a^{2} {(b^{2})}^{2} c} \cdot \sqrt{3 a^{6} b^{2} c}$

Step 1.2

Move $6$ .

$\sqrt{a^{2} {(b^{2})}^{2} \cdot 6 c} \cdot \sqrt{3 a^{6} b^{2} c}$

Step 1.3

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

$\sqrt{{(a b^{2})}^{2} \cdot 6 c} \cdot \sqrt{3 a^{6} b^{2} c}$

Step 1.4

Add parentheses.

$\sqrt{{(a b^{2})}^{2} \cdot (6 c)} \cdot \sqrt{3 a^{6} b^{2} c}$

Step 2

Pull terms out from under the radical.

$a b^{2} \sqrt{6 c} \cdot \sqrt{3 a^{6} b^{2} c}$

Step 3

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

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Step 3.1

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

$a b^{2} \sqrt{6 c} \cdot \sqrt{3 {(a^{3})}^{2} b^{2} c}$

Step 3.2

Move $3$ .

$a b^{2} \sqrt{6 c} \cdot \sqrt{{(a^{3})}^{2} b^{2} \cdot 3 c}$

Step 3.3

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

$a b^{2} \sqrt{6 c} \cdot \sqrt{{(a^{3} b)}^{2} \cdot 3 c}$

Step 3.4

Add parentheses.

$a b^{2} \sqrt{6 c} \cdot \sqrt{{(a^{3} b)}^{2} \cdot (3 c)}$

Step 4

Pull terms out from under the radical.

$a b^{2} \sqrt{6 c} \cdot (a^{3} b \sqrt{3 c})$

Step 5

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

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Step 5.1

Move $a^{3}$ .

$a^{3} a b^{2} \sqrt{6 c} \cdot (b \sqrt{3 c})$

Step 5.2

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

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Step 5.2.1

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

$a^{3} a^{1} b^{2} \sqrt{6 c} \cdot (b \sqrt{3 c})$

Step 5.2.2

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

$a^{3 + 1} b^{2} \sqrt{6 c} \cdot (b \sqrt{3 c})$

Step 5.3

Add $3$ and $1$ .

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

Step 6

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

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Step 6.1

Move $b$ .

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

Step 6.2

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

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Step 6.2.1

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

$a^{4} (b^{1} b^{2}) \sqrt{6 c} \cdot \sqrt{3 c}$

Step 6.2.2

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

$a^{4} b^{1 + 2} \sqrt{6 c} \cdot \sqrt{3 c}$

Step 6.3

Add $1$ and $2$ .

$a^{4} b^{3} \sqrt{6 c} \cdot \sqrt{3 c}$

Step 7

Multiply $a^{4} b^{3} \sqrt{6 c} \sqrt{3 c}$ .

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Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

Multiply $6$ by $3$ .

$a^{4} b^{3} \sqrt{18 c \cdot c}$

Step 7.3

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

$a^{4} b^{3} \sqrt{18 (c^{1} c)}$

Step 7.4

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

$a^{4} b^{3} \sqrt{18 (c^{1} c^{1})}$

Step 7.5

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

$a^{4} b^{3} \sqrt{18 c^{1 + 1}}$

Step 7.6

Add $1$ and $1$ .

$a^{4} b^{3} \sqrt{18 c^{2}}$

Step 8

Rewrite $18 c^{2}$ as ${(3 c)}^{2} \cdot 2$ .

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Step 8.1

Factor $9$ out of $18$ .

$a^{4} b^{3} \sqrt{9 (2) c^{2}}$

Step 8.2

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

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

Step 8.3

Move $2$ .

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

Step 8.4

Rewrite $3^{2} c^{2}$ as ${(3 c)}^{2}$ .

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

Step 9

Pull terms out from under the radical.

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

Step 10

Rewrite using the commutative property of multiplication.

$3 \sqrt{2} a^{4} b^{3} c$