Pre-Algebra Examples

$\sqrt{192 x} \cdot \sqrt{3 x y^{7}}$

Step 1

Rewrite $192 x$ as $8^{2} \cdot (3 x)$ .

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Step 1.1

Factor $64$ out of $192$ .

$\sqrt{64 (3) x} \cdot \sqrt{3 x y^{7}}$

Step 1.2

Rewrite $64$ as $8^{2}$ .

$\sqrt{8^{2} \cdot 3 x} \cdot \sqrt{3 x y^{7}}$

Step 1.3

Add parentheses.

$\sqrt{8^{2} \cdot (3 x)} \cdot \sqrt{3 x y^{7}}$

Step 2

Pull terms out from under the radical.

$8 \sqrt{3 x} \cdot \sqrt{3 x y^{7}}$

Step 3

Rewrite $3 x y^{7}$ as ${(y^{3})}^{2} \cdot (3 x y)$ .

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Step 3.1

Factor out $y^{6}$ .

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

Step 3.2

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

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

Step 3.3

Move $x$ .

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

Step 3.4

Reorder $3$ and ${(y^{3})}^{2}$ .

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

Step 3.5

Add parentheses.

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

Step 3.6

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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Step 5.1

Combine using the product rule for radicals.

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

Step 5.2

Multiply $3$ by $3$ .

$8 (y^{3} \sqrt{9 x (x y)})$

Step 5.3

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

$8 (y^{3} \sqrt{9 (x^{1} x) y})$

Step 5.4

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

$8 (y^{3} \sqrt{9 (x^{1} x^{1}) y})$

Step 5.5

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

$8 (y^{3} \sqrt{9 x^{1 + 1} y})$

Step 5.6

Add $1$ and $1$ .

$8 (y^{3} \sqrt{9 x^{2} y})$

Step 6

Rewrite $9 x^{2} y$ as ${(3 x)}^{2} y$ .

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Pull terms out from under the radical.

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

Step 8

Simplify the expression.

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Step 8.1

Rewrite using the commutative property of multiplication.

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

Step 8.2

Multiply $3$ by $8$ .

$24 (y^{3} (x \sqrt{y}))$

Step 8.3

Reorder factors in $24 y^{3} (x \sqrt{y})$ .

$24 y^{3} x \sqrt{y}$