Algebra Examples

$\frac{1}{2} \cdot 10 x (7 x^{3} + 5 x) - \frac{1}{2} \cdot 3 x (4 x^{3} + 2 x)$

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 $10 x (7 x^{3} + 5 x)$ .

$\frac{1}{2} \cdot (2 (5 x (7 x^{3} + 5 x))) - \frac{1}{2} \cdot (3 x (4 x^{3} + 2 x))$

Step 1.1.2

Cancel the common factor.

$\frac{1}{2} \cdot (2 (5 x (7 x^{3} + 5 x))) - \frac{1}{2} \cdot (3 x (4 x^{3} + 2 x))$

Step 1.1.3

Rewrite the expression.

$5 x (7 x^{3} + 5 x) - \frac{1}{2} \cdot (3 x (4 x^{3} + 2 x))$

Step 1.2

Apply the distributive property.

$5 x (7 x^{3}) + 5 x (5 x) - \frac{1}{2} \cdot (3 x (4 x^{3} + 2 x))$

Step 1.3

Rewrite using the commutative property of multiplication.

$5 \cdot 7 x \cdot x^{3} + 5 x (5 x) - \frac{1}{2} \cdot (3 x (4 x^{3} + 2 x))$

Step 1.4

Rewrite using the commutative property of multiplication.

$5 \cdot 7 x \cdot x^{3} + 5 \cdot 5 x \cdot x - \frac{1}{2} \cdot (3 x (4 x^{3} + 2 x))$

Step 1.5

Simplify each term.

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Step 1.5.1

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

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Step 1.5.1.1

Move $x^{3}$ .

$5 \cdot 7 (x^{3} x) + 5 \cdot 5 x \cdot x - \frac{1}{2} \cdot (3 x (4 x^{3} + 2 x))$

Step 1.5.1.2

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

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Step 1.5.1.2.1

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

$5 \cdot 7 (x^{3} x^{1}) + 5 \cdot 5 x \cdot x - \frac{1}{2} \cdot (3 x (4 x^{3} + 2 x))$

Step 1.5.1.2.2

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

$5 \cdot 7 x^{3 + 1} + 5 \cdot 5 x \cdot x - \frac{1}{2} \cdot (3 x (4 x^{3} + 2 x))$

Step 1.5.1.3

Add $3$ and $1$ .

$5 \cdot 7 x^{4} + 5 \cdot 5 x \cdot x - \frac{1}{2} \cdot (3 x (4 x^{3} + 2 x))$

Step 1.5.2

Multiply $5$ by $7$ .

$35 x^{4} + 5 \cdot 5 x \cdot x - \frac{1}{2} \cdot (3 x (4 x^{3} + 2 x))$

Step 1.5.3

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

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Step 1.5.3.1

Move $x$ .

$35 x^{4} + 5 \cdot 5 (x \cdot x) - \frac{1}{2} \cdot (3 x (4 x^{3} + 2 x))$

Step 1.5.3.2

Multiply $x$ by $x$ .

$35 x^{4} + 5 \cdot 5 x^{2} - \frac{1}{2} \cdot (3 x (4 x^{3} + 2 x))$

Step 1.5.4

Multiply $5$ by $5$ .

$35 x^{4} + 25 x^{2} - \frac{1}{2} \cdot (3 x (4 x^{3} + 2 x))$

Step 1.6

Apply the distributive property.

$35 x^{4} + 25 x^{2} - \frac{1}{2} \cdot (3 x (4 x^{3}) + 3 x (2 x))$

Step 1.7

Rewrite using the commutative property of multiplication.

$35 x^{4} + 25 x^{2} - \frac{1}{2} \cdot (3 \cdot 4 x \cdot x^{3} + 3 x (2 x))$

Step 1.8

Rewrite using the commutative property of multiplication.

$35 x^{4} + 25 x^{2} - \frac{1}{2} \cdot (3 \cdot 4 x \cdot x^{3} + 3 \cdot 2 x \cdot x)$

Step 1.9

Simplify each term.

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Step 1.9.1

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

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Step 1.9.1.1

Move $x^{3}$ .

$35 x^{4} + 25 x^{2} - \frac{1}{2} \cdot (3 \cdot 4 (x^{3} x) + 3 \cdot 2 x \cdot x)$

Step 1.9.1.2

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

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Step 1.9.1.2.1

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

$35 x^{4} + 25 x^{2} - \frac{1}{2} \cdot (3 \cdot 4 (x^{3} x^{1}) + 3 \cdot 2 x \cdot x)$

Step 1.9.1.2.2

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

$35 x^{4} + 25 x^{2} - \frac{1}{2} \cdot (3 \cdot 4 x^{3 + 1} + 3 \cdot 2 x \cdot x)$

Step 1.9.1.3

Add $3$ and $1$ .

$35 x^{4} + 25 x^{2} - \frac{1}{2} \cdot (3 \cdot 4 x^{4} + 3 \cdot 2 x \cdot x)$

Step 1.9.2

Multiply $3$ by $4$ .

$35 x^{4} + 25 x^{2} - \frac{1}{2} \cdot (12 x^{4} + 3 \cdot 2 x \cdot x)$

Step 1.9.3

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

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Step 1.9.3.1

Move $x$ .

$35 x^{4} + 25 x^{2} - \frac{1}{2} \cdot (12 x^{4} + 3 \cdot 2 (x \cdot x))$

Step 1.9.3.2

Multiply $x$ by $x$ .

$35 x^{4} + 25 x^{2} - \frac{1}{2} \cdot (12 x^{4} + 3 \cdot 2 x^{2})$

Step 1.9.4

Multiply $3$ by $2$ .

$35 x^{4} + 25 x^{2} - \frac{1}{2} \cdot (12 x^{4} + 6 x^{2})$

Step 1.10

Apply the distributive property.

$35 x^{4} + 25 x^{2} - \frac{1}{2} (12 x^{4}) - \frac{1}{2} (6 x^{2})$

Step 1.11

Cancel the common factor of $2$ .

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Step 1.11.1

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

$35 x^{4} + 25 x^{2} + \frac{- 1}{2} (12 x^{4}) - \frac{1}{2} (6 x^{2})$

Step 1.11.2

Factor $2$ out of $12 x^{4}$ .

$35 x^{4} + 25 x^{2} + \frac{- 1}{2} (2 (6 x^{4})) - \frac{1}{2} (6 x^{2})$

Step 1.11.3

Cancel the common factor.

$35 x^{4} + 25 x^{2} + \frac{- 1}{2} (2 (6 x^{4})) - \frac{1}{2} (6 x^{2})$

Step 1.11.4

Rewrite the expression.

$35 x^{4} + 25 x^{2} - 1 (6 x^{4}) - \frac{1}{2} (6 x^{2})$

Step 1.12

Multiply $6$ by $- 1$ .

$35 x^{4} + 25 x^{2} - 6 x^{4} - \frac{1}{2} (6 x^{2})$

Step 1.13

Cancel the common factor of $2$ .

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Step 1.13.1

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

$35 x^{4} + 25 x^{2} - 6 x^{4} + \frac{- 1}{2} (6 x^{2})$

Step 1.13.2

Factor $2$ out of $6 x^{2}$ .

$35 x^{4} + 25 x^{2} - 6 x^{4} + \frac{- 1}{2} (2 (3 x^{2}))$

Step 1.13.3

Cancel the common factor.

$35 x^{4} + 25 x^{2} - 6 x^{4} + \frac{- 1}{2} (2 (3 x^{2}))$

Step 1.13.4

Rewrite the expression.

$35 x^{4} + 25 x^{2} - 6 x^{4} - 1 (3 x^{2})$

Step 1.14

Multiply $3$ by $- 1$ .

$35 x^{4} + 25 x^{2} - 6 x^{4} - 3 x^{2}$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $6 x^{4}$ from $35 x^{4}$ .

$29 x^{4} + 25 x^{2} - 3 x^{2}$

Step 2.2

Subtract $3 x^{2}$ from $25 x^{2}$ .

$29 x^{4} + 22 x^{2}$