Algebra Examples

$\sqrt{2 a} (2 \sqrt{8 a} + \sqrt{2 a^{3}})$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $4$ out of $8$ .

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

Step 1.1.2

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

$\sqrt{2 a} (2 \sqrt{2^{2} \cdot 2 a} + \sqrt{2 a^{3}})$

Step 1.1.3

Add parentheses.

$\sqrt{2 a} (2 \sqrt{2^{2} \cdot (2 a)} + \sqrt{2 a^{3}})$

Step 1.2

Pull terms out from under the radical.

$\sqrt{2 a} (2 (2 \sqrt{2 a}) + \sqrt{2 a^{3}})$

Step 1.3

Multiply $2$ by $2$ .

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

Step 1.4

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

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Step 1.4.1

Factor out $a^{2}$ .

$\sqrt{2 a} (4 \sqrt{2 a} + \sqrt{2 (a^{2} a)})$

Step 1.4.2

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

$\sqrt{2 a} (4 \sqrt{2 a} + \sqrt{a^{2} \cdot 2 a})$

Step 1.4.3

Add parentheses.

$\sqrt{2 a} (4 \sqrt{2 a} + \sqrt{a^{2} \cdot (2 a)})$

Step 1.5

Pull terms out from under the radical.

$\sqrt{2 a} (4 \sqrt{2 a} + a \sqrt{2 a})$

Step 2

Simplify by multiplying through.

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Step 2.1

Apply the distributive property.

$\sqrt{2 a} (4 \sqrt{2 a}) + \sqrt{2 a} (a \sqrt{2 a})$

Step 2.2

Rewrite using the commutative property of multiplication.

$4 \sqrt{2 a} \sqrt{2 a} + \sqrt{2 a} (a \sqrt{2 a})$

Step 3

Multiply $\sqrt{2 a} (a \sqrt{2 a})$ .

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Step 3.1

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

$4 \sqrt{2 a} \sqrt{2 a} + a ({\sqrt{2 a}}^{1} \sqrt{2 a})$

Step 3.2

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

$4 \sqrt{2 a} \sqrt{2 a} + a ({\sqrt{2 a}}^{1} {\sqrt{2 a}}^{1})$

Step 3.3

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

$4 \sqrt{2 a} \sqrt{2 a} + a {\sqrt{2 a}}^{1 + 1}$

Step 3.4

Add $1$ and $1$ .

$4 \sqrt{2 a} \sqrt{2 a} + a {\sqrt{2 a}}^{2}$

Step 4

Simplify each term.

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Step 4.1

Multiply $4 \sqrt{2 a} \sqrt{2 a}$ .

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Step 4.1.1

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

$4 ({\sqrt{2 a}}^{1} \sqrt{2 a}) + a {\sqrt{2 a}}^{2}$

Step 4.1.2

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

$4 ({\sqrt{2 a}}^{1} {\sqrt{2 a}}^{1}) + a {\sqrt{2 a}}^{2}$

Step 4.1.3

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

$4 {\sqrt{2 a}}^{1 + 1} + a {\sqrt{2 a}}^{2}$

Step 4.1.4

Add $1$ and $1$ .

$4 {\sqrt{2 a}}^{2} + a {\sqrt{2 a}}^{2}$

Step 4.2

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

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Step 4.2.1

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

$4 {({(2 a)}^{\frac{1}{2}})}^{2} + a {\sqrt{2 a}}^{2}$

Step 4.2.2

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

$4 {(2 a)}^{\frac{1}{2} \cdot 2} + a {\sqrt{2 a}}^{2}$

Step 4.2.3

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

$4 {(2 a)}^{\frac{2}{2}} + a {\sqrt{2 a}}^{2}$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

$4 {(2 a)}^{\frac{2}{2}} + a {\sqrt{2 a}}^{2}$

Step 4.2.4.2

Rewrite the expression.

$4 {(2 a)}^{1} + a {\sqrt{2 a}}^{2}$

Step 4.2.5

Simplify.

$4 (2 a) + a {\sqrt{2 a}}^{2}$

Step 4.3

Multiply $2$ by $4$ .

$8 a + a {\sqrt{2 a}}^{2}$

Step 4.4

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

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Step 4.4.1

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

$8 a + a {({(2 a)}^{\frac{1}{2}})}^{2}$

Step 4.4.2

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

$8 a + a {(2 a)}^{\frac{1}{2} \cdot 2}$

Step 4.4.3

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

$8 a + a {(2 a)}^{\frac{2}{2}}$

Step 4.4.4

Cancel the common factor of $2$ .

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Step 4.4.4.1

Cancel the common factor.

$8 a + a {(2 a)}^{\frac{2}{2}}$

Step 4.4.4.2

Rewrite the expression.

$8 a + a {(2 a)}^{1}$

Step 4.4.5

Simplify.

$8 a + a (2 a)$

Step 4.5

Rewrite using the commutative property of multiplication.

$8 a + 2 a \cdot a$

Step 4.6

Multiply $a$ by $a$ by adding the exponents.

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Step 4.6.1

Move $a$ .

$8 a + 2 (a \cdot a)$

Step 4.6.2

Multiply $a$ by $a$ .

$8 a + 2 a^{2}$