Algebra Examples

$2 \sqrt{2 x} (3 \sqrt{6 x^{3}} - 4 \sqrt{8 y} + 7 \sqrt{x y^{2}})$

Step 1

Simplify each term.

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Step 1.1

Rewrite $6 x^{3}$ as $x^{2} \cdot (6 x)$ .

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Step 1.1.1

Factor out $x^{2}$ .

$2 \sqrt{2 x} (3 \sqrt{6 (x^{2} x)} - 4 \sqrt{8 y} + 7 \sqrt{x y^{2}})$

Step 1.1.2

Reorder $6$ and $x^{2}$ .

$2 \sqrt{2 x} (3 \sqrt{x^{2} \cdot 6 x} - 4 \sqrt{8 y} + 7 \sqrt{x y^{2}})$

Step 1.1.3

Add parentheses.

$2 \sqrt{2 x} (3 \sqrt{x^{2} \cdot (6 x)} - 4 \sqrt{8 y} + 7 \sqrt{x y^{2}})$

Step 1.2

Pull terms out from under the radical.

$2 \sqrt{2 x} (3 (x \sqrt{6 x}) - 4 \sqrt{8 y} + 7 \sqrt{x y^{2}})$

Step 1.3

Rewrite $8 y$ as $2^{2} \cdot (2 y)$ .

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Step 1.3.1

Factor $4$ out of $8$ .

$2 \sqrt{2 x} (3 x \sqrt{6 x} - 4 \sqrt{4 (2) y} + 7 \sqrt{x y^{2}})$

Step 1.3.2

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

$2 \sqrt{2 x} (3 x \sqrt{6 x} - 4 \sqrt{2^{2} \cdot 2 y} + 7 \sqrt{x y^{2}})$

Step 1.3.3

Add parentheses.

$2 \sqrt{2 x} (3 x \sqrt{6 x} - 4 \sqrt{2^{2} \cdot (2 y)} + 7 \sqrt{x y^{2}})$

Step 1.4

Pull terms out from under the radical.

$2 \sqrt{2 x} (3 x \sqrt{6 x} - 4 (2 \sqrt{2 y}) + 7 \sqrt{x y^{2}})$

Step 1.5

Multiply $2$ by $- 4$ .

$2 \sqrt{2 x} (3 x \sqrt{6 x} - 8 \sqrt{2 y} + 7 \sqrt{x y^{2}})$

Step 1.6

Reorder $x$ and $y^{2}$ .

$2 \sqrt{2 x} (3 x \sqrt{6 x} - 8 \sqrt{2 y} + 7 \sqrt{y^{2} x})$

Step 1.7

Pull terms out from under the radical.

$2 \sqrt{2 x} (3 x \sqrt{6 x} - 8 \sqrt{2 y} + 7 y \sqrt{x})$

Step 2

Apply the distributive property.

$2 \sqrt{2 x} (3 x \sqrt{6 x}) + 2 \sqrt{2 x} (- 8 \sqrt{2 y}) + 2 \sqrt{2 x} (7 y \sqrt{x})$

Step 3

Simplify.

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Step 3.1

Multiply $2 \sqrt{2 x} (3 x \sqrt{6 x})$ .

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Step 3.1.1

Multiply $3$ by $2$ .

$6 \sqrt{2 x} (x \sqrt{6 x}) + 2 \sqrt{2 x} (- 8 \sqrt{2 y}) + 2 \sqrt{2 x} (7 y \sqrt{x})$

Step 3.1.2

Combine using the product rule for radicals.

$6 (x \sqrt{2 x (6 x)}) + 2 \sqrt{2 x} (- 8 \sqrt{2 y}) + 2 \sqrt{2 x} (7 y \sqrt{x})$

Step 3.1.3

Multiply $6$ by $2$ .

$6 (x \sqrt{12 x \cdot x}) + 2 \sqrt{2 x} (- 8 \sqrt{2 y}) + 2 \sqrt{2 x} (7 y \sqrt{x})$

Step 3.1.4

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

$6 (x \sqrt{12 (x^{1} x)}) + 2 \sqrt{2 x} (- 8 \sqrt{2 y}) + 2 \sqrt{2 x} (7 y \sqrt{x})$

Step 3.1.5

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

$6 (x \sqrt{12 (x^{1} x^{1})}) + 2 \sqrt{2 x} (- 8 \sqrt{2 y}) + 2 \sqrt{2 x} (7 y \sqrt{x})$

Step 3.1.6

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

$6 (x \sqrt{12 x^{1 + 1}}) + 2 \sqrt{2 x} (- 8 \sqrt{2 y}) + 2 \sqrt{2 x} (7 y \sqrt{x})$

Step 3.1.7

Add $1$ and $1$ .

$6 (x \sqrt{12 x^{2}}) + 2 \sqrt{2 x} (- 8 \sqrt{2 y}) + 2 \sqrt{2 x} (7 y \sqrt{x})$

Step 3.2

Multiply $2 \sqrt{2 x} (- 8 \sqrt{2 y})$ .

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Step 3.2.1

Multiply $- 8$ by $2$ .

$6 (x \sqrt{12 x^{2}}) - 16 \sqrt{2 x} \sqrt{2 y} + 2 \sqrt{2 x} (7 y \sqrt{x})$

Step 3.2.2

Combine using the product rule for radicals.

$6 (x \sqrt{12 x^{2}}) - 16 \sqrt{2 y (2 x)} + 2 \sqrt{2 x} (7 y \sqrt{x})$

Step 3.2.3

Multiply $2$ by $2$ .

$6 (x \sqrt{12 x^{2}}) - 16 \sqrt{4 y x} + 2 \sqrt{2 x} (7 y \sqrt{x})$

Step 3.3

Multiply $2 \sqrt{2 x} (7 y \sqrt{x})$ .

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Step 3.3.1

Multiply $7$ by $2$ .

$6 (x \sqrt{12 x^{2}}) - 16 \sqrt{4 y x} + 14 \sqrt{2 x} (y \sqrt{x})$

Step 3.3.2

Combine using the product rule for radicals.

$6 (x \sqrt{12 x^{2}}) - 16 \sqrt{4 y x} + 14 (y \sqrt{2 x \cdot x})$

Step 3.3.3

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

$6 (x \sqrt{12 x^{2}}) - 16 \sqrt{4 y x} + 14 (y \sqrt{2 (x^{1} x)})$

Step 3.3.4

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

$6 (x \sqrt{12 x^{2}}) - 16 \sqrt{4 y x} + 14 (y \sqrt{2 (x^{1} x^{1})})$

Step 3.3.5

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

$6 (x \sqrt{12 x^{2}}) - 16 \sqrt{4 y x} + 14 (y \sqrt{2 x^{1 + 1}})$

Step 3.3.6

Add $1$ and $1$ .

$6 (x \sqrt{12 x^{2}}) - 16 \sqrt{4 y x} + 14 (y \sqrt{2 x^{2}})$

Step 4

Simplify each term.

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Step 4.1

Rewrite $12 x^{2}$ as ${(2 x)}^{2} \cdot 3$ .

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Step 4.1.1

Factor $4$ out of $12$ .

$6 (x \sqrt{4 (3) x^{2}}) - 16 \sqrt{4 y x} + 14 (y \sqrt{2 x^{2}})$

Step 4.1.2

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

$6 (x \sqrt{2^{2} \cdot 3 x^{2}}) - 16 \sqrt{4 y x} + 14 (y \sqrt{2 x^{2}})$

Step 4.1.3

Move $3$ .

$6 (x \sqrt{2^{2} x^{2} \cdot 3}) - 16 \sqrt{4 y x} + 14 (y \sqrt{2 x^{2}})$

Step 4.1.4

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

$6 (x \sqrt{{(2 x)}^{2} \cdot 3}) - 16 \sqrt{4 y x} + 14 (y \sqrt{2 x^{2}})$

Step 4.2

Pull terms out from under the radical.

$6 (x (2 x \sqrt{3})) - 16 \sqrt{4 y x} + 14 (y \sqrt{2 x^{2}})$

Step 4.3

Multiply $x$ by $x$ by adding the exponents.

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Step 4.3.1

Move $x$ .

$6 (x \cdot x (2 \sqrt{3})) - 16 \sqrt{4 y x} + 14 (y \sqrt{2 x^{2}})$

Step 4.3.2

Multiply $x$ by $x$ .

$6 (x^{2} (2 \sqrt{3})) - 16 \sqrt{4 y x} + 14 (y \sqrt{2 x^{2}})$

Step 4.4

Move $2$ to the left of $x^{2}$ .

$6 (2 \cdot x^{2} \sqrt{3}) - 16 \sqrt{4 y x} + 14 (y \sqrt{2 x^{2}})$

Step 4.5

Multiply $2$ by $6$ .

$12 (x^{2} \sqrt{3}) - 16 \sqrt{4 y x} + 14 (y \sqrt{2 x^{2}})$

Step 4.6

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

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Step 4.6.1

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

$12 x^{2} \sqrt{3} - 16 \sqrt{2^{2} y x} + 14 (y \sqrt{2 x^{2}})$

Step 4.6.2

Add parentheses.

$12 x^{2} \sqrt{3} - 16 \sqrt{2^{2} (y x)} + 14 (y \sqrt{2 x^{2}})$

Step 4.7

Pull terms out from under the radical.

$12 x^{2} \sqrt{3} - 16 (2 \sqrt{y x}) + 14 (y \sqrt{2 x^{2}})$

Step 4.8

Multiply $2$ by $- 16$ .

$12 x^{2} \sqrt{3} - 32 \sqrt{y x} + 14 (y \sqrt{2 x^{2}})$

Step 4.9

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

$12 x^{2} \sqrt{3} - 32 \sqrt{y x} + 14 (y \sqrt{x^{2} \cdot 2})$

Step 4.10

Pull terms out from under the radical.

$12 x^{2} \sqrt{3} - 32 \sqrt{y x} + 14 (y (x \sqrt{2}))$

Step 4.11

Rewrite using the commutative property of multiplication.

$12 x^{2} \sqrt{3} - 32 \sqrt{y x} + 14 \sqrt{2} y x$

Step 5

Simplify the expression.

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Step 5.1

Reorder $y$ and $x$ .

$12 x^{2} \sqrt{3} - 32 \sqrt{x y} + 14 \sqrt{2} y x$

Step 5.2

Move $- 32 \sqrt{x y}$ .

$12 x^{2} \sqrt{3} + 14 \sqrt{2} y x - 32 \sqrt{x y}$