Algebra Examples

$(3 + \sqrt[3]{5}) (9 - 3 \sqrt[3]{5} + \sqrt[3]{25})$

Step 1

Expand $(3 + \sqrt[3]{5}) (9 - 3 \sqrt[3]{5} + \sqrt[3]{25})$ by multiplying each term in the first expression by each term in the second expression.

$3 \cdot 9 + 3 (- 3 \sqrt[3]{5}) + 3 \sqrt[3]{25} + \sqrt[3]{5} \cdot 9 + \sqrt[3]{5} (- 3 \sqrt[3]{5}) + \sqrt[3]{5} \sqrt[3]{25}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $3$ by $9$ .

$27 + 3 (- 3 \sqrt[3]{5}) + 3 \sqrt[3]{25} + \sqrt[3]{5} \cdot 9 + \sqrt[3]{5} (- 3 \sqrt[3]{5}) + \sqrt[3]{5} \sqrt[3]{25}$

Step 2.1.2

Multiply $- 3$ by $3$ .

$27 - 9 \sqrt[3]{5} + 3 \sqrt[3]{25} + \sqrt[3]{5} \cdot 9 + \sqrt[3]{5} (- 3 \sqrt[3]{5}) + \sqrt[3]{5} \sqrt[3]{25}$

Step 2.1.3

Move $9$ to the left of $\sqrt[3]{5}$ .

$27 - 9 \sqrt[3]{5} + 3 \sqrt[3]{25} + 9 \cdot \sqrt[3]{5} + \sqrt[3]{5} (- 3 \sqrt[3]{5}) + \sqrt[3]{5} \sqrt[3]{25}$

Step 2.1.4

Multiply $\sqrt[3]{5} (- 3 \sqrt[3]{5})$ .

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Step 2.1.4.1

Raise $\sqrt[3]{5}$ to the power of $1$ .

$27 - 9 \sqrt[3]{5} + 3 \sqrt[3]{25} + 9 \sqrt[3]{5} - 3 ({\sqrt[3]{5}}^{1} \sqrt[3]{5}) + \sqrt[3]{5} \sqrt[3]{25}$

Step 2.1.4.2

Raise $\sqrt[3]{5}$ to the power of $1$ .

$27 - 9 \sqrt[3]{5} + 3 \sqrt[3]{25} + 9 \sqrt[3]{5} - 3 ({\sqrt[3]{5}}^{1} {\sqrt[3]{5}}^{1}) + \sqrt[3]{5} \sqrt[3]{25}$

Step 2.1.4.3

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

$27 - 9 \sqrt[3]{5} + 3 \sqrt[3]{25} + 9 \sqrt[3]{5} - 3 {\sqrt[3]{5}}^{1 + 1} + \sqrt[3]{5} \sqrt[3]{25}$

Step 2.1.4.4

Add $1$ and $1$ .

$27 - 9 \sqrt[3]{5} + 3 \sqrt[3]{25} + 9 \sqrt[3]{5} - 3 {\sqrt[3]{5}}^{2} + \sqrt[3]{5} \sqrt[3]{25}$

Step 2.1.5

Rewrite ${\sqrt[3]{5}}^{2}$ as $\sqrt[3]{5^{2}}$ .

$27 - 9 \sqrt[3]{5} + 3 \sqrt[3]{25} + 9 \sqrt[3]{5} - 3 \sqrt[3]{5^{2}} + \sqrt[3]{5} \sqrt[3]{25}$

Step 2.1.6

Raise $5$ to the power of $2$ .

$27 - 9 \sqrt[3]{5} + 3 \sqrt[3]{25} + 9 \sqrt[3]{5} - 3 \sqrt[3]{25} + \sqrt[3]{5} \sqrt[3]{25}$

Step 2.1.7

Combine using the product rule for radicals.

$27 - 9 \sqrt[3]{5} + 3 \sqrt[3]{25} + 9 \sqrt[3]{5} - 3 \sqrt[3]{25} + \sqrt[3]{5 \cdot 25}$

Step 2.1.8

Multiply $5$ by $25$ .

$27 - 9 \sqrt[3]{5} + 3 \sqrt[3]{25} + 9 \sqrt[3]{5} - 3 \sqrt[3]{25} + \sqrt[3]{125}$

Step 2.1.9

Rewrite $125$ as $5^{3}$ .

$27 - 9 \sqrt[3]{5} + 3 \sqrt[3]{25} + 9 \sqrt[3]{5} - 3 \sqrt[3]{25} + \sqrt[3]{5^{3}}$

Step 2.1.10

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

$27 - 9 \sqrt[3]{5} + 3 \sqrt[3]{25} + 9 \sqrt[3]{5} - 3 \sqrt[3]{25} + 5$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $27$ and $5$ .

$32 - 9 \sqrt[3]{5} + 3 \sqrt[3]{25} + 9 \sqrt[3]{5} - 3 \sqrt[3]{25}$

Step 2.2.2

Add $- 9 \sqrt[3]{5}$ and $9 \sqrt[3]{5}$ .

$32 + 0 + 3 \sqrt[3]{25} - 3 \sqrt[3]{25}$

Step 2.2.3

Add $32$ and $0$ .

$32 + 3 \sqrt[3]{25} - 3 \sqrt[3]{25}$

Step 2.2.4

Subtract $3 \sqrt[3]{25}$ from $3 \sqrt[3]{25}$ .

$32 + 0$

Step 2.2.5

Add $32$ and $0$ .

$32$