Pre-Algebra Examples

$(3 + \sqrt[3]{9}) (9 - 3 \sqrt[3]{9} + \sqrt[3]{81})$

Step 1

Simplify each term.

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Step 1.1

Rewrite $81$ as $3^{3} \cdot 3$ .

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Step 1.1.1

Factor $27$ out of $81$ .

$(3 + \sqrt[3]{9}) (9 - 3 \sqrt[3]{9} + \sqrt[3]{27 (3)})$

Step 1.1.2

Rewrite $27$ as $3^{3}$ .

$(3 + \sqrt[3]{9}) (9 - 3 \sqrt[3]{9} + \sqrt[3]{3^{3} \cdot 3})$

Step 1.2

Pull terms out from under the radical.

$(3 + \sqrt[3]{9}) (9 - 3 \sqrt[3]{9} + 3 \sqrt[3]{3})$

Step 2

Expand $(3 + \sqrt[3]{9}) (9 - 3 \sqrt[3]{9} + 3 \sqrt[3]{3})$ by multiplying each term in the first expression by each term in the second expression.

$3 \cdot 9 + 3 (- 3 \sqrt[3]{9}) + 3 (3 \sqrt[3]{3}) + \sqrt[3]{9} \cdot 9 + \sqrt[3]{9} (- 3 \sqrt[3]{9}) + \sqrt[3]{9} (3 \sqrt[3]{3})$

Step 3

Simplify each term.

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Step 3.1

Multiply $3$ by $9$ .

$27 + 3 (- 3 \sqrt[3]{9}) + 3 (3 \sqrt[3]{3}) + \sqrt[3]{9} \cdot 9 + \sqrt[3]{9} (- 3 \sqrt[3]{9}) + \sqrt[3]{9} (3 \sqrt[3]{3})$

Step 3.2

Multiply $- 3$ by $3$ .

$27 - 9 \sqrt[3]{9} + 3 (3 \sqrt[3]{3}) + \sqrt[3]{9} \cdot 9 + \sqrt[3]{9} (- 3 \sqrt[3]{9}) + \sqrt[3]{9} (3 \sqrt[3]{3})$

Step 3.3

Multiply $3$ by $3$ .

$27 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + \sqrt[3]{9} \cdot 9 + \sqrt[3]{9} (- 3 \sqrt[3]{9}) + \sqrt[3]{9} (3 \sqrt[3]{3})$

Step 3.4

Move $9$ to the left of $\sqrt[3]{9}$ .

$27 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + 9 \cdot \sqrt[3]{9} + \sqrt[3]{9} (- 3 \sqrt[3]{9}) + \sqrt[3]{9} (3 \sqrt[3]{3})$

Step 3.5

Multiply $\sqrt[3]{9} (- 3 \sqrt[3]{9})$ .

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Step 3.5.1

Raise $\sqrt[3]{9}$ to the power of $1$ .

$27 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + 9 \sqrt[3]{9} - 3 ({\sqrt[3]{9}}^{1} \sqrt[3]{9}) + \sqrt[3]{9} (3 \sqrt[3]{3})$

Step 3.5.2

Raise $\sqrt[3]{9}$ to the power of $1$ .

$27 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + 9 \sqrt[3]{9} - 3 ({\sqrt[3]{9}}^{1} {\sqrt[3]{9}}^{1}) + \sqrt[3]{9} (3 \sqrt[3]{3})$

Step 3.5.3

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

$27 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + 9 \sqrt[3]{9} - 3 {\sqrt[3]{9}}^{1 + 1} + \sqrt[3]{9} (3 \sqrt[3]{3})$

Step 3.5.4

Add $1$ and $1$ .

$27 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + 9 \sqrt[3]{9} - 3 {\sqrt[3]{9}}^{2} + \sqrt[3]{9} (3 \sqrt[3]{3})$

Step 3.6

Rewrite ${\sqrt[3]{9}}^{2}$ as $\sqrt[3]{9^{2}}$ .

$27 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + 9 \sqrt[3]{9} - 3 \sqrt[3]{9^{2}} + \sqrt[3]{9} (3 \sqrt[3]{3})$

Step 3.7

Raise $9$ to the power of $2$ .

$27 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + 9 \sqrt[3]{9} - 3 \sqrt[3]{81} + \sqrt[3]{9} (3 \sqrt[3]{3})$

Step 3.8

Rewrite $81$ as $3^{3} \cdot 3$ .

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Step 3.8.1

Factor $27$ out of $81$ .

$27 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + 9 \sqrt[3]{9} - 3 \sqrt[3]{27 (3)} + \sqrt[3]{9} (3 \sqrt[3]{3})$

Step 3.8.2

Rewrite $27$ as $3^{3}$ .

$27 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + 9 \sqrt[3]{9} - 3 \sqrt[3]{3^{3} \cdot 3} + \sqrt[3]{9} (3 \sqrt[3]{3})$

Step 3.9

Pull terms out from under the radical.

$27 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + 9 \sqrt[3]{9} - 3 (3 \sqrt[3]{3}) + \sqrt[3]{9} (3 \sqrt[3]{3})$

Step 3.10

Multiply $3$ by $- 3$ .

$27 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + 9 \sqrt[3]{9} - 9 \sqrt[3]{3} + \sqrt[3]{9} (3 \sqrt[3]{3})$

Step 3.11

Multiply $\sqrt[3]{9} (3 \sqrt[3]{3})$ .

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Step 3.11.1

Combine using the product rule for radicals.

$27 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + 9 \sqrt[3]{9} - 9 \sqrt[3]{3} + 3 \sqrt[3]{9 \cdot 3}$

Step 3.11.2

Multiply $9$ by $3$ .

$27 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + 9 \sqrt[3]{9} - 9 \sqrt[3]{3} + 3 \sqrt[3]{27}$

Step 3.12

Rewrite $27$ as $3^{3}$ .

$27 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + 9 \sqrt[3]{9} - 9 \sqrt[3]{3} + 3 \sqrt[3]{3^{3}}$

Step 3.13

Pull terms out from under the radical, assuming real numbers.

$27 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + 9 \sqrt[3]{9} - 9 \sqrt[3]{3} + 3 \cdot 3$

Step 3.14

Multiply $3$ by $3$ .

$27 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + 9 \sqrt[3]{9} - 9 \sqrt[3]{3} + 9$

Step 4

Add $27$ and $9$ .

$36 - 9 \sqrt[3]{9} + 9 \sqrt[3]{3} + 9 \sqrt[3]{9} - 9 \sqrt[3]{3}$

Step 5

Add $- 9 \sqrt[3]{9}$ and $9 \sqrt[3]{9}$ .

$36 + 0 + 9 \sqrt[3]{3} - 9 \sqrt[3]{3}$

Step 6

Add $36$ and $0$ .

$36 + 9 \sqrt[3]{3} - 9 \sqrt[3]{3}$

Step 7

Subtract $9 \sqrt[3]{3}$ from $9 \sqrt[3]{3}$ .

$36 + 0$

Step 8

Add $36$ and $0$ .

$36$