Algebra Examples

$\sqrt{2 x} (\sqrt{8 x} - \sqrt{32})$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $4$ out of $8$ .

$\sqrt{2 x} (\sqrt{4 (2) x} - \sqrt{32})$

Step 1.1.2

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

$\sqrt{2 x} (\sqrt{2^{2} \cdot 2 x} - \sqrt{32})$

Step 1.1.3

Add parentheses.

$\sqrt{2 x} (\sqrt{2^{2} \cdot (2 x)} - \sqrt{32})$

Step 1.2

Pull terms out from under the radical.

$\sqrt{2 x} (2 \sqrt{2 x} - \sqrt{32})$

Step 1.3

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 1.3.1

Factor $16$ out of $32$ .

$\sqrt{2 x} (2 \sqrt{2 x} - \sqrt{16 (2)})$

Step 1.3.2

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

$\sqrt{2 x} (2 \sqrt{2 x} - \sqrt{4^{2} \cdot 2})$

Step 1.4

Pull terms out from under the radical.

$\sqrt{2 x} (2 \sqrt{2 x} - (4 \sqrt{2}))$

Step 1.5

Multiply $4$ by $- 1$ .

$\sqrt{2 x} (2 \sqrt{2 x} - 4 \sqrt{2})$

Step 2

Simplify by multiplying through.

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Step 2.1

Apply the distributive property.

$\sqrt{2 x} (2 \sqrt{2 x}) + \sqrt{2 x} (- 4 \sqrt{2})$

Step 2.2

Rewrite using the commutative property of multiplication.

$2 \sqrt{2 x} \sqrt{2 x} + \sqrt{2 x} (- 4 \sqrt{2})$

Step 3

Multiply $\sqrt{2 x} (- 4 \sqrt{2})$ .

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Step 3.1

Combine using the product rule for radicals.

$2 \sqrt{2 x} \sqrt{2 x} - 4 \sqrt{2 x \cdot 2}$

Step 3.2

Multiply $2$ by $2$ .

$2 \sqrt{2 x} \sqrt{2 x} - 4 \sqrt{4 x}$

Step 4

Simplify each term.

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Step 4.1

Multiply $2 \sqrt{2 x} \sqrt{2 x}$ .

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Step 4.1.1

Raise $\sqrt{2 x}$ to the power of $1$ .

$2 ({\sqrt{2 x}}^{1} \sqrt{2 x}) - 4 \sqrt{4 x}$

Step 4.1.2

Raise $\sqrt{2 x}$ to the power of $1$ .

$2 ({\sqrt{2 x}}^{1} {\sqrt{2 x}}^{1}) - 4 \sqrt{4 x}$

Step 4.1.3

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

$2 {\sqrt{2 x}}^{1 + 1} - 4 \sqrt{4 x}$

Step 4.1.4

Add $1$ and $1$ .

$2 {\sqrt{2 x}}^{2} - 4 \sqrt{4 x}$

Step 4.2

Rewrite ${\sqrt{2 x}}^{2}$ as $2 x$ .

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x}$ as ${(2 x)}^{\frac{1}{2}}$ .

$2 {({(2 x)}^{\frac{1}{2}})}^{2} - 4 \sqrt{4 x}$

Step 4.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 {(2 x)}^{\frac{1}{2} \cdot 2} - 4 \sqrt{4 x}$

Step 4.2.3

Combine $\frac{1}{2}$ and $2$ .

$2 {(2 x)}^{\frac{2}{2}} - 4 \sqrt{4 x}$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

$2 {(2 x)}^{\frac{2}{2}} - 4 \sqrt{4 x}$

Step 4.2.4.2

Rewrite the expression.

$2 {(2 x)}^{1} - 4 \sqrt{4 x}$

Step 4.2.5

Simplify.

$2 (2 x) - 4 \sqrt{4 x}$

Step 4.3

Multiply $2$ by $2$ .

$4 x - 4 \sqrt{4 x}$

Step 4.4

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

$4 x - 4 \sqrt{2^{2} x}$

Step 4.5

Pull terms out from under the radical.

$4 x - 4 (2 \sqrt{x})$

Step 4.6

Multiply $2$ by $- 4$ .

$4 x - 8 \sqrt{x}$