Algebra Examples

${(3 x^{2} + x)}^{3}$

Step 1

Use the Binomial Theorem.

${(3 x^{2})}^{3} + 3 {(3 x^{2})}^{2} x + 3 (3 x^{2}) x^{2} + x^{3}$

Step 2

Simplify each term.

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Step 2.1

Apply the product rule to $3 x^{2}$ .

$3^{3} {(x^{2})}^{3} + 3 {(3 x^{2})}^{2} x + 3 (3 x^{2}) x^{2} + x^{3}$

Step 2.2

Raise $3$ to the power of $3$ .

$27 {(x^{2})}^{3} + 3 {(3 x^{2})}^{2} x + 3 (3 x^{2}) x^{2} + x^{3}$

Step 2.3

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 2.3.1

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

$27 x^{2 \cdot 3} + 3 {(3 x^{2})}^{2} x + 3 (3 x^{2}) x^{2} + x^{3}$

Step 2.3.2

Multiply $2$ by $3$ .

$27 x^{6} + 3 {(3 x^{2})}^{2} x + 3 (3 x^{2}) x^{2} + x^{3}$

Step 2.4

Apply the product rule to $3 x^{2}$ .

$27 x^{6} + 3 (3^{2} {(x^{2})}^{2}) x + 3 (3 x^{2}) x^{2} + x^{3}$

Step 2.5

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

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Step 2.5.1

Move $3^{2}$ .

$27 x^{6} + 3^{2} \cdot 3 {(x^{2})}^{2} x + 3 (3 x^{2}) x^{2} + x^{3}$

Step 2.5.2

Multiply $3^{2}$ by $3$ .

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Step 2.5.2.1

Raise $3$ to the power of $1$ .

$27 x^{6} + 3^{2} \cdot 3^{1} {(x^{2})}^{2} x + 3 (3 x^{2}) x^{2} + x^{3}$

Step 2.5.2.2

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

$27 x^{6} + 3^{2 + 1} {(x^{2})}^{2} x + 3 (3 x^{2}) x^{2} + x^{3}$

Step 2.5.3

Add $2$ and $1$ .

$27 x^{6} + 3^{3} {(x^{2})}^{2} x + 3 (3 x^{2}) x^{2} + x^{3}$

Step 2.6

Raise $3$ to the power of $3$ .

$27 x^{6} + 27 {(x^{2})}^{2} x + 3 (3 x^{2}) x^{2} + x^{3}$

Step 2.7

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 2.7.1

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

$27 x^{6} + 27 x^{2 \cdot 2} x + 3 (3 x^{2}) x^{2} + x^{3}$

Step 2.7.2

Multiply $2$ by $2$ .

$27 x^{6} + 27 x^{4} x + 3 (3 x^{2}) x^{2} + x^{3}$

Step 2.8

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

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Step 2.8.1

Move $x$ .

$27 x^{6} + 27 (x \cdot x^{4}) + 3 (3 x^{2}) x^{2} + x^{3}$

Step 2.8.2

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

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Step 2.8.2.1

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

$27 x^{6} + 27 (x^{1} x^{4}) + 3 (3 x^{2}) x^{2} + x^{3}$

Step 2.8.2.2

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

$27 x^{6} + 27 x^{1 + 4} + 3 (3 x^{2}) x^{2} + x^{3}$

Step 2.8.3

Add $1$ and $4$ .

$27 x^{6} + 27 x^{5} + 3 (3 x^{2}) x^{2} + x^{3}$

Step 2.9

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

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Step 2.9.1

Move $x^{2}$ .

$27 x^{6} + 27 x^{5} + 3 (3 (x^{2} x^{2})) + x^{3}$

Step 2.9.2

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

$27 x^{6} + 27 x^{5} + 3 (3 x^{2 + 2}) + x^{3}$

Step 2.9.3

Add $2$ and $2$ .

$27 x^{6} + 27 x^{5} + 3 (3 x^{4}) + x^{3}$

Step 2.10

Multiply $3$ by $3$ .

$27 x^{6} + 27 x^{5} + 9 x^{4} + x^{3}$