Algebra Examples

${(\frac{1}{2} x^{3} + 6 x)}^{2}$

Step 1

Combine fractions.

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Step 1.1

Combine $\frac{1}{2}$ and $x^{3}$ .

${(\frac{x^{3}}{2} + 6 x)}^{2}$

Step 1.2

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

$(\frac{x^{3}}{2} + 6 x) (\frac{x^{3}}{2} + 6 x)$

Step 2

Expand $(\frac{x^{3}}{2} + 6 x) (\frac{x^{3}}{2} + 6 x)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{x^{3}}{2} (\frac{x^{3}}{2} + 6 x) + 6 x (\frac{x^{3}}{2} + 6 x)$

Step 2.2

Apply the distributive property.

$\frac{x^{3}}{2} \cdot \frac{x^{3}}{2} + \frac{x^{3}}{2} (6 x) + 6 x (\frac{x^{3}}{2} + 6 x)$

Step 2.3

Apply the distributive property.

$\frac{x^{3}}{2} \cdot \frac{x^{3}}{2} + \frac{x^{3}}{2} (6 x) + 6 x \frac{x^{3}}{2} + 6 x (6 x)$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Combine.

$\frac{x^{3} x^{3}}{2 \cdot 2} + \frac{x^{3}}{2} (6 x) + 6 x \frac{x^{3}}{2} + 6 x (6 x)$

Step 3.1.2

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

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Step 3.1.2.1

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

$\frac{x^{3 + 3}}{2 \cdot 2} + \frac{x^{3}}{2} (6 x) + 6 x \frac{x^{3}}{2} + 6 x (6 x)$

Step 3.1.2.2

Add $3$ and $3$ .

$\frac{x^{6}}{2 \cdot 2} + \frac{x^{3}}{2} (6 x) + 6 x \frac{x^{3}}{2} + 6 x (6 x)$

Step 3.1.3

Multiply $2$ by $2$ .

$\frac{x^{6}}{4} + \frac{x^{3}}{2} (6 x) + 6 x \frac{x^{3}}{2} + 6 x (6 x)$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$\frac{x^{6}}{4} + 6 \frac{x^{3}}{2} x + 6 x \frac{x^{3}}{2} + 6 x (6 x)$

Step 3.1.5

Cancel the common factor of $2$ .

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Step 3.1.5.1

Factor $2$ out of $6$ .

$\frac{x^{6}}{4} + 2 (3) \frac{x^{3}}{2} x + 6 x \frac{x^{3}}{2} + 6 x (6 x)$

Step 3.1.5.2

Cancel the common factor.

$\frac{x^{6}}{4} + 2 \cdot 3 \frac{x^{3}}{2} x + 6 x \frac{x^{3}}{2} + 6 x (6 x)$

Step 3.1.5.3

Rewrite the expression.

$\frac{x^{6}}{4} + 3 x^{3} x + 6 x \frac{x^{3}}{2} + 6 x (6 x)$

Step 3.1.6

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 3.1.6.1

Move $x$ .

$\frac{x^{6}}{4} + 3 (x \cdot x^{3}) + 6 x \frac{x^{3}}{2} + 6 x (6 x)$

Step 3.1.6.2

Multiply $x$ by $x^{3}$ .

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Step 3.1.6.2.1

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

$\frac{x^{6}}{4} + 3 (x^{1} x^{3}) + 6 x \frac{x^{3}}{2} + 6 x (6 x)$

Step 3.1.6.2.2

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

$\frac{x^{6}}{4} + 3 x^{1 + 3} + 6 x \frac{x^{3}}{2} + 6 x (6 x)$

Step 3.1.6.3

Add $1$ and $3$ .

$\frac{x^{6}}{4} + 3 x^{4} + 6 x \frac{x^{3}}{2} + 6 x (6 x)$

Step 3.1.7

Cancel the common factor of $2$ .

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Step 3.1.7.1

Factor $2$ out of $6 x$ .

$\frac{x^{6}}{4} + 3 x^{4} + 2 (3 x) \frac{x^{3}}{2} + 6 x (6 x)$

Step 3.1.7.2

Cancel the common factor.

$\frac{x^{6}}{4} + 3 x^{4} + 2 (3 x) \frac{x^{3}}{2} + 6 x (6 x)$

Step 3.1.7.3

Rewrite the expression.

$\frac{x^{6}}{4} + 3 x^{4} + 3 x \cdot x^{3} + 6 x (6 x)$

Step 3.1.8

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 3.1.8.1

Move $x^{3}$ .

$\frac{x^{6}}{4} + 3 x^{4} + 3 (x^{3} x) + 6 x (6 x)$

Step 3.1.8.2

Multiply $x^{3}$ by $x$ .

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Step 3.1.8.2.1

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

$\frac{x^{6}}{4} + 3 x^{4} + 3 (x^{3} x^{1}) + 6 x (6 x)$

Step 3.1.8.2.2

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

$\frac{x^{6}}{4} + 3 x^{4} + 3 x^{3 + 1} + 6 x (6 x)$

Step 3.1.8.3

Add $3$ and $1$ .

$\frac{x^{6}}{4} + 3 x^{4} + 3 x^{4} + 6 x (6 x)$

Step 3.1.9

Rewrite using the commutative property of multiplication.

$\frac{x^{6}}{4} + 3 x^{4} + 3 x^{4} + 6 \cdot 6 x \cdot x$

Step 3.1.10

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

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Step 3.1.10.1

Move $x$ .

$\frac{x^{6}}{4} + 3 x^{4} + 3 x^{4} + 6 \cdot 6 (x \cdot x)$

Step 3.1.10.2

Multiply $x$ by $x$ .

$\frac{x^{6}}{4} + 3 x^{4} + 3 x^{4} + 6 \cdot 6 x^{2}$

Step 3.1.11

Multiply $6$ by $6$ .

$\frac{x^{6}}{4} + 3 x^{4} + 3 x^{4} + 36 x^{2}$

Step 3.2

Add $3 x^{4}$ and $3 x^{4}$ .

$\frac{x^{6}}{4} + 6 x^{4} + 36 x^{2}$