Algebra Examples

$2 x \sqrt{3 x y} \cdot 4 \sqrt{2 x^{5} y}$

Step 1

Multiply $4$ by $2$ .

$8 x \sqrt{3 x y} \sqrt{2 x^{5} y}$

Step 2

Rewrite $2 x^{5} y$ as ${(x^{2})}^{2} \cdot (2 x y)$ .

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Step 2.1

Factor out $x^{4}$ .

$8 x \sqrt{3 x y} \sqrt{2 (x^{4} x) y}$

Step 2.2

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

$8 x \sqrt{3 x y} \sqrt{2 ({(x^{2})}^{2} x) y}$

Step 2.3

Reorder $2$ and ${(x^{2})}^{2}$ .

$8 x \sqrt{3 x y} \sqrt{{(x^{2})}^{2} \cdot 2 x y}$

Step 2.4

Add parentheses.

$8 x \sqrt{3 x y} \sqrt{{(x^{2})}^{2} \cdot 2 (x y)}$

Step 2.5

Add parentheses.

$8 x \sqrt{3 x y} \sqrt{{(x^{2})}^{2} \cdot (2 x y)}$

Step 3

Pull terms out from under the radical.

$8 x \sqrt{3 x y} (| x^{2} | \sqrt{2 x y})$

Step 4

Remove non-negative terms from the absolute value.

$8 x \sqrt{3 x y} (x^{2} \sqrt{2 x y})$

Step 5

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

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Step 5.1

Move $x^{2}$ .

$8 (x^{2} x) \sqrt{3 x y} \sqrt{2 x y}$

Step 5.2

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

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Step 5.2.1

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

$8 (x^{2} x^{1}) \sqrt{3 x y} \sqrt{2 x y}$

Step 5.2.2

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

$8 x^{2 + 1} \sqrt{3 x y} \sqrt{2 x y}$

Step 5.3

Add $2$ and $1$ .

$8 x^{3} \sqrt{3 x y} \sqrt{2 x y}$

Step 6

Multiply $8 x^{3} \sqrt{3 x y} \sqrt{2 x y}$ .

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Step 6.1

Combine using the product rule for radicals.

$8 x^{3} \sqrt{2 x y (3 x y)}$

Step 6.2

Multiply $3$ by $2$ .

$8 x^{3} \sqrt{6 x y (x y)}$

Step 6.3

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

$8 x^{3} \sqrt{6 (x^{1} x) y \cdot y}$

Step 6.4

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

$8 x^{3} \sqrt{6 (x^{1} x^{1}) y \cdot y}$

Step 6.5

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

$8 x^{3} \sqrt{6 x^{1 + 1} y \cdot y}$

Step 6.6

Add $1$ and $1$ .

$8 x^{3} \sqrt{6 x^{2} y \cdot y}$

Step 6.7

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

$8 x^{3} \sqrt{6 x^{2} (y^{1} y)}$

Step 6.8

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

$8 x^{3} \sqrt{6 x^{2} (y^{1} y^{1})}$

Step 6.9

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

$8 x^{3} \sqrt{6 x^{2} y^{1 + 1}}$

Step 6.10

Add $1$ and $1$ .

$8 x^{3} \sqrt{6 x^{2} y^{2}}$

Step 7

Rewrite $6 x^{2} y^{2}$ as ${(x y)}^{2} \cdot 6$ .

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Step 7.1

Move $6$ .

$8 x^{3} \sqrt{x^{2} y^{2} \cdot 6}$

Step 7.2

Rewrite $x^{2} y^{2}$ as ${(x y)}^{2}$ .

$8 x^{3} \sqrt{{(x y)}^{2} \cdot 6}$

Step 8

Pull terms out from under the radical.

$8 x^{3} (| x y | \sqrt{6})$

Step 9

Reorder factors in $8 x^{3} (| x y | \sqrt{6})$ .

$8 x^{3} | x y | \sqrt{6}$