Algebra Examples

$\frac{1}{2} \cdot 4 x^{2} (8 x^{3} + x^{2} + 2 x) - \frac{1}{2} \cdot 2 x^{2} (6 x^{3} + 2 x)$

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $2$ .

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Step 1.1.1

Factor $2$ out of $4 x^{2} (8 x^{3} + x^{2} + 2 x)$ .

$\frac{1}{2} \cdot (2 (2 x^{2} (8 x^{3} + x^{2} + 2 x))) - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.1.2

Cancel the common factor.

$\frac{1}{2} \cdot (2 (2 x^{2} (8 x^{3} + x^{2} + 2 x))) - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.1.3

Rewrite the expression.

$2 x^{2} (8 x^{3} + x^{2} + 2 x) - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.2

Apply the distributive property.

$2 x^{2} (8 x^{3}) + 2 x^{2} x^{2} + 2 x^{2} (2 x) - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

$2 \cdot 8 x^{2} x^{3} + 2 x^{2} x^{2} + 2 x^{2} (2 x) - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.3.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.3.2.1

Move $x^{2}$ .

$2 \cdot 8 x^{2} x^{3} + 2 (x^{2} x^{2}) + 2 x^{2} (2 x) - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.3.2.2

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

$2 \cdot 8 x^{2} x^{3} + 2 x^{2 + 2} + 2 x^{2} (2 x) - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.3.2.3

Add $2$ and $2$ .

$2 \cdot 8 x^{2} x^{3} + 2 x^{4} + 2 x^{2} (2 x) - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.3.3

Rewrite using the commutative property of multiplication.

$2 \cdot 8 x^{2} x^{3} + 2 x^{4} + 2 \cdot 2 x^{2} x - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.4

Simplify each term.

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Step 1.4.1

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 1.4.1.1

Move $x^{3}$ .

$2 \cdot 8 (x^{3} x^{2}) + 2 x^{4} + 2 \cdot 2 x^{2} x - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.4.1.2

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

$2 \cdot 8 x^{3 + 2} + 2 x^{4} + 2 \cdot 2 x^{2} x - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.4.1.3

Add $3$ and $2$ .

$2 \cdot 8 x^{5} + 2 x^{4} + 2 \cdot 2 x^{2} x - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.4.2

Multiply $2$ by $8$ .

$16 x^{5} + 2 x^{4} + 2 \cdot 2 x^{2} x - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.4.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.4.3.1

Move $x$ .

$16 x^{5} + 2 x^{4} + 2 \cdot 2 (x \cdot x^{2}) - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.4.3.2

Multiply $x$ by $x^{2}$ .

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Step 1.4.3.2.1

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

$16 x^{5} + 2 x^{4} + 2 \cdot 2 (x^{1} x^{2}) - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.4.3.2.2

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

$16 x^{5} + 2 x^{4} + 2 \cdot 2 x^{1 + 2} - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.4.3.3

Add $1$ and $2$ .

$16 x^{5} + 2 x^{4} + 2 \cdot 2 x^{3} - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.4.4

Multiply $2$ by $2$ .

$16 x^{5} + 2 x^{4} + 4 x^{3} - \frac{1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$16 x^{5} + 2 x^{4} + 4 x^{3} + \frac{- 1}{2} \cdot (2 x^{2} (6 x^{3} + 2 x))$

Step 1.5.2

Factor $2$ out of $2 x^{2} (6 x^{3} + 2 x)$ .

$16 x^{5} + 2 x^{4} + 4 x^{3} + \frac{- 1}{2} \cdot (2 (x^{2} (6 x^{3} + 2 x)))$

Step 1.5.3

Cancel the common factor.

$16 x^{5} + 2 x^{4} + 4 x^{3} + \frac{- 1}{2} \cdot (2 (x^{2} (6 x^{3} + 2 x)))$

Step 1.5.4

Rewrite the expression.

$16 x^{5} + 2 x^{4} + 4 x^{3} - 1 \cdot (x^{2} (6 x^{3} + 2 x))$

Step 1.6

Apply the distributive property.

$16 x^{5} + 2 x^{4} + 4 x^{3} - 1 \cdot (x^{2} (6 x^{3}) + x^{2} (2 x))$

Step 1.7

Rewrite using the commutative property of multiplication.

$16 x^{5} + 2 x^{4} + 4 x^{3} - 1 \cdot (6 x^{2} x^{3} + x^{2} (2 x))$

Step 1.8

Rewrite using the commutative property of multiplication.

$16 x^{5} + 2 x^{4} + 4 x^{3} - 1 \cdot (6 x^{2} x^{3} + 2 x^{2} x)$

Step 1.9

Simplify each term.

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Step 1.9.1

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 1.9.1.1

Move $x^{3}$ .

$16 x^{5} + 2 x^{4} + 4 x^{3} - 1 \cdot (6 (x^{3} x^{2}) + 2 x^{2} x)$

Step 1.9.1.2

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

$16 x^{5} + 2 x^{4} + 4 x^{3} - 1 \cdot (6 x^{3 + 2} + 2 x^{2} x)$

Step 1.9.1.3

Add $3$ and $2$ .

$16 x^{5} + 2 x^{4} + 4 x^{3} - 1 \cdot (6 x^{5} + 2 x^{2} x)$

Step 1.9.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.9.2.1

Move $x$ .

$16 x^{5} + 2 x^{4} + 4 x^{3} - 1 \cdot (6 x^{5} + 2 (x \cdot x^{2}))$

Step 1.9.2.2

Multiply $x$ by $x^{2}$ .

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Step 1.9.2.2.1

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

$16 x^{5} + 2 x^{4} + 4 x^{3} - 1 \cdot (6 x^{5} + 2 (x^{1} x^{2}))$

Step 1.9.2.2.2

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

$16 x^{5} + 2 x^{4} + 4 x^{3} - 1 \cdot (6 x^{5} + 2 x^{1 + 2})$

Step 1.9.2.3

Add $1$ and $2$ .

$16 x^{5} + 2 x^{4} + 4 x^{3} - 1 \cdot (6 x^{5} + 2 x^{3})$

Step 1.10

Apply the distributive property.

$16 x^{5} + 2 x^{4} + 4 x^{3} - 1 (6 x^{5}) - 1 (2 x^{3})$

Step 1.11

Multiply $6$ by $- 1$ .

$16 x^{5} + 2 x^{4} + 4 x^{3} - 6 x^{5} - 1 (2 x^{3})$

Step 1.12

Multiply $2$ by $- 1$ .

$16 x^{5} + 2 x^{4} + 4 x^{3} - 6 x^{5} - 2 x^{3}$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $6 x^{5}$ from $16 x^{5}$ .

$10 x^{5} + 2 x^{4} + 4 x^{3} - 2 x^{3}$

Step 2.2

Subtract $2 x^{3}$ from $4 x^{3}$ .

$10 x^{5} + 2 x^{4} + 2 x^{3}$