Algebra Examples

$9 x^{5} + 3 x {(4 x^{4} - 3 x^{2})}^{2}$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(4 x^{4} - 3 x^{2})}^{2}$ as $(4 x^{4} - 3 x^{2}) (4 x^{4} - 3 x^{2})$ .

$9 x^{5} + 3 x ((4 x^{4} - 3 x^{2}) (4 x^{4} - 3 x^{2}))$

Step 1.2

Expand $(4 x^{4} - 3 x^{2}) (4 x^{4} - 3 x^{2})$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$9 x^{5} + 3 x (4 x^{4} (4 x^{4} - 3 x^{2}) - 3 x^{2} (4 x^{4} - 3 x^{2}))$

Step 1.2.2

Apply the distributive property.

$9 x^{5} + 3 x (4 x^{4} (4 x^{4}) + 4 x^{4} (- 3 x^{2}) - 3 x^{2} (4 x^{4} - 3 x^{2}))$

Step 1.2.3

Apply the distributive property.

$9 x^{5} + 3 x (4 x^{4} (4 x^{4}) + 4 x^{4} (- 3 x^{2}) - 3 x^{2} (4 x^{4}) - 3 x^{2} (- 3 x^{2}))$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

$9 x^{5} + 3 x (4 \cdot 4 x^{4} x^{4} + 4 x^{4} (- 3 x^{2}) - 3 x^{2} (4 x^{4}) - 3 x^{2} (- 3 x^{2}))$

Step 1.3.1.2

Multiply $x^{4}$ by $x^{4}$ by adding the exponents.

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Step 1.3.1.2.1

Move $x^{4}$ .

$9 x^{5} + 3 x (4 \cdot 4 (x^{4} x^{4}) + 4 x^{4} (- 3 x^{2}) - 3 x^{2} (4 x^{4}) - 3 x^{2} (- 3 x^{2}))$

Step 1.3.1.2.2

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

$9 x^{5} + 3 x (4 \cdot 4 x^{4 + 4} + 4 x^{4} (- 3 x^{2}) - 3 x^{2} (4 x^{4}) - 3 x^{2} (- 3 x^{2}))$

Step 1.3.1.2.3

Add $4$ and $4$ .

$9 x^{5} + 3 x (4 \cdot 4 x^{8} + 4 x^{4} (- 3 x^{2}) - 3 x^{2} (4 x^{4}) - 3 x^{2} (- 3 x^{2}))$

Step 1.3.1.3

Multiply $4$ by $4$ .

$9 x^{5} + 3 x (16 x^{8} + 4 x^{4} (- 3 x^{2}) - 3 x^{2} (4 x^{4}) - 3 x^{2} (- 3 x^{2}))$

Step 1.3.1.4

Rewrite using the commutative property of multiplication.

$9 x^{5} + 3 x (16 x^{8} + 4 \cdot - 3 x^{4} x^{2} - 3 x^{2} (4 x^{4}) - 3 x^{2} (- 3 x^{2}))$

Step 1.3.1.5

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 1.3.1.5.1

Move $x^{2}$ .

$9 x^{5} + 3 x (16 x^{8} + 4 \cdot - 3 (x^{2} x^{4}) - 3 x^{2} (4 x^{4}) - 3 x^{2} (- 3 x^{2}))$

Step 1.3.1.5.2

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

$9 x^{5} + 3 x (16 x^{8} + 4 \cdot - 3 x^{2 + 4} - 3 x^{2} (4 x^{4}) - 3 x^{2} (- 3 x^{2}))$

Step 1.3.1.5.3

Add $2$ and $4$ .

$9 x^{5} + 3 x (16 x^{8} + 4 \cdot - 3 x^{6} - 3 x^{2} (4 x^{4}) - 3 x^{2} (- 3 x^{2}))$

Step 1.3.1.6

Multiply $4$ by $- 3$ .

$9 x^{5} + 3 x (16 x^{8} - 12 x^{6} - 3 x^{2} (4 x^{4}) - 3 x^{2} (- 3 x^{2}))$

Step 1.3.1.7

Rewrite using the commutative property of multiplication.

$9 x^{5} + 3 x (16 x^{8} - 12 x^{6} - 3 \cdot 4 x^{2} x^{4} - 3 x^{2} (- 3 x^{2}))$

Step 1.3.1.8

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

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Step 1.3.1.8.1

Move $x^{4}$ .

$9 x^{5} + 3 x (16 x^{8} - 12 x^{6} - 3 \cdot 4 (x^{4} x^{2}) - 3 x^{2} (- 3 x^{2}))$

Step 1.3.1.8.2

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

$9 x^{5} + 3 x (16 x^{8} - 12 x^{6} - 3 \cdot 4 x^{4 + 2} - 3 x^{2} (- 3 x^{2}))$

Step 1.3.1.8.3

Add $4$ and $2$ .

$9 x^{5} + 3 x (16 x^{8} - 12 x^{6} - 3 \cdot 4 x^{6} - 3 x^{2} (- 3 x^{2}))$

Step 1.3.1.9

Multiply $- 3$ by $4$ .

$9 x^{5} + 3 x (16 x^{8} - 12 x^{6} - 12 x^{6} - 3 x^{2} (- 3 x^{2}))$

Step 1.3.1.10

Rewrite using the commutative property of multiplication.

$9 x^{5} + 3 x (16 x^{8} - 12 x^{6} - 12 x^{6} - 3 \cdot - 3 x^{2} x^{2})$

Step 1.3.1.11

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

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Step 1.3.1.11.1

Move $x^{2}$ .

$9 x^{5} + 3 x (16 x^{8} - 12 x^{6} - 12 x^{6} - 3 \cdot - 3 (x^{2} x^{2}))$

Step 1.3.1.11.2

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

$9 x^{5} + 3 x (16 x^{8} - 12 x^{6} - 12 x^{6} - 3 \cdot - 3 x^{2 + 2})$

Step 1.3.1.11.3

Add $2$ and $2$ .

$9 x^{5} + 3 x (16 x^{8} - 12 x^{6} - 12 x^{6} - 3 \cdot - 3 x^{4})$

Step 1.3.1.12

Multiply $- 3$ by $- 3$ .

$9 x^{5} + 3 x (16 x^{8} - 12 x^{6} - 12 x^{6} + 9 x^{4})$

Step 1.3.2

Subtract $12 x^{6}$ from $- 12 x^{6}$ .

$9 x^{5} + 3 x (16 x^{8} - 24 x^{6} + 9 x^{4})$

Step 1.4

Apply the distributive property.

$9 x^{5} + 3 x (16 x^{8}) + 3 x (- 24 x^{6}) + 3 x (9 x^{4})$

Step 1.5

Simplify.

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Step 1.5.1

Rewrite using the commutative property of multiplication.

$9 x^{5} + 3 \cdot 16 x \cdot x^{8} + 3 x (- 24 x^{6}) + 3 x (9 x^{4})$

Step 1.5.2

Rewrite using the commutative property of multiplication.

$9 x^{5} + 3 \cdot 16 x \cdot x^{8} + 3 \cdot - 24 x \cdot x^{6} + 3 x (9 x^{4})$

Step 1.5.3

Rewrite using the commutative property of multiplication.

$9 x^{5} + 3 \cdot 16 x \cdot x^{8} + 3 \cdot - 24 x \cdot x^{6} + 3 \cdot 9 x \cdot x^{4}$

Step 1.6

Simplify each term.

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Step 1.6.1

Multiply $x$ by $x^{8}$ by adding the exponents.

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Step 1.6.1.1

Move $x^{8}$ .

$9 x^{5} + 3 \cdot 16 (x^{8} x) + 3 \cdot - 24 x \cdot x^{6} + 3 \cdot 9 x \cdot x^{4}$

Step 1.6.1.2

Multiply $x^{8}$ by $x$ .

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Step 1.6.1.2.1

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

$9 x^{5} + 3 \cdot 16 (x^{8} x^{1}) + 3 \cdot - 24 x \cdot x^{6} + 3 \cdot 9 x \cdot x^{4}$

Step 1.6.1.2.2

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

$9 x^{5} + 3 \cdot 16 x^{8 + 1} + 3 \cdot - 24 x \cdot x^{6} + 3 \cdot 9 x \cdot x^{4}$

Step 1.6.1.3

Add $8$ and $1$ .

$9 x^{5} + 3 \cdot 16 x^{9} + 3 \cdot - 24 x \cdot x^{6} + 3 \cdot 9 x \cdot x^{4}$

Step 1.6.2

Multiply $3$ by $16$ .

$9 x^{5} + 48 x^{9} + 3 \cdot - 24 x \cdot x^{6} + 3 \cdot 9 x \cdot x^{4}$

Step 1.6.3

Multiply $x$ by $x^{6}$ by adding the exponents.

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Step 1.6.3.1

Move $x^{6}$ .

$9 x^{5} + 48 x^{9} + 3 \cdot - 24 (x^{6} x) + 3 \cdot 9 x \cdot x^{4}$

Step 1.6.3.2

Multiply $x^{6}$ by $x$ .

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Step 1.6.3.2.1

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

$9 x^{5} + 48 x^{9} + 3 \cdot - 24 (x^{6} x^{1}) + 3 \cdot 9 x \cdot x^{4}$

Step 1.6.3.2.2

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

$9 x^{5} + 48 x^{9} + 3 \cdot - 24 x^{6 + 1} + 3 \cdot 9 x \cdot x^{4}$

Step 1.6.3.3

Add $6$ and $1$ .

$9 x^{5} + 48 x^{9} + 3 \cdot - 24 x^{7} + 3 \cdot 9 x \cdot x^{4}$

Step 1.6.4

Multiply $3$ by $- 24$ .

$9 x^{5} + 48 x^{9} - 72 x^{7} + 3 \cdot 9 x \cdot x^{4}$

Step 1.6.5

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 1.6.5.1

Move $x^{4}$ .

$9 x^{5} + 48 x^{9} - 72 x^{7} + 3 \cdot 9 (x^{4} x)$

Step 1.6.5.2

Multiply $x^{4}$ by $x$ .

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Step 1.6.5.2.1

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

$9 x^{5} + 48 x^{9} - 72 x^{7} + 3 \cdot 9 (x^{4} x^{1})$

Step 1.6.5.2.2

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

$9 x^{5} + 48 x^{9} - 72 x^{7} + 3 \cdot 9 x^{4 + 1}$

Step 1.6.5.3

Add $4$ and $1$ .

$9 x^{5} + 48 x^{9} - 72 x^{7} + 3 \cdot 9 x^{5}$

Step 1.6.6

Multiply $3$ by $9$ .

$9 x^{5} + 48 x^{9} - 72 x^{7} + 27 x^{5}$

Step 2

Add $9 x^{5}$ and $27 x^{5}$ .

$48 x^{9} - 72 x^{7} + 36 x^{5}$