Pre-Algebra Examples

$(\sqrt[3]{x} + 5) (\sqrt[3]{x^{2}} - 5 \sqrt[3]{x} + 25)$

Step 1

Expand $(\sqrt[3]{x} + 5) (\sqrt[3]{x^{2}} - 5 \sqrt[3]{x} + 25)$ by multiplying each term in the first expression by each term in the second expression.

$\sqrt[3]{x} \sqrt[3]{x^{2}} + \sqrt[3]{x} (- 5 \sqrt[3]{x}) + \sqrt[3]{x} \cdot 25 + 5 \sqrt[3]{x^{2}} + 5 (- 5 \sqrt[3]{x}) + 5 \cdot 25$

Step 2

Simplify each term.

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Step 2.1

Multiply $\sqrt[3]{x} \sqrt[3]{x^{2}}$ .

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Step 2.1.1

Combine using the product rule for radicals.

$\sqrt[3]{x \cdot x^{2}} + \sqrt[3]{x} (- 5 \sqrt[3]{x}) + \sqrt[3]{x} \cdot 25 + 5 \sqrt[3]{x^{2}} + 5 (- 5 \sqrt[3]{x}) + 5 \cdot 25$

Step 2.1.2

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

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Step 2.1.2.1

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

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Step 2.1.2.1.1

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

$\sqrt[3]{x^{1} x^{2}} + \sqrt[3]{x} (- 5 \sqrt[3]{x}) + \sqrt[3]{x} \cdot 25 + 5 \sqrt[3]{x^{2}} + 5 (- 5 \sqrt[3]{x}) + 5 \cdot 25$

Step 2.1.2.1.2

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

$\sqrt[3]{x^{1 + 2}} + \sqrt[3]{x} (- 5 \sqrt[3]{x}) + \sqrt[3]{x} \cdot 25 + 5 \sqrt[3]{x^{2}} + 5 (- 5 \sqrt[3]{x}) + 5 \cdot 25$

Step 2.1.2.2

Add $1$ and $2$ .

$\sqrt[3]{x^{3}} + \sqrt[3]{x} (- 5 \sqrt[3]{x}) + \sqrt[3]{x} \cdot 25 + 5 \sqrt[3]{x^{2}} + 5 (- 5 \sqrt[3]{x}) + 5 \cdot 25$

Step 2.2

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

$x + \sqrt[3]{x} (- 5 \sqrt[3]{x}) + \sqrt[3]{x} \cdot 25 + 5 \sqrt[3]{x^{2}} + 5 (- 5 \sqrt[3]{x}) + 5 \cdot 25$

Step 2.3

Rewrite using the commutative property of multiplication.

$x - 5 \sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot 25 + 5 \sqrt[3]{x^{2}} + 5 (- 5 \sqrt[3]{x}) + 5 \cdot 25$

Step 2.4

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

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Step 2.4.1

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

$x - 5 ({\sqrt[3]{x}}^{1} \sqrt[3]{x}) + \sqrt[3]{x} \cdot 25 + 5 \sqrt[3]{x^{2}} + 5 (- 5 \sqrt[3]{x}) + 5 \cdot 25$

Step 2.4.2

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

$x - 5 ({\sqrt[3]{x}}^{1} {\sqrt[3]{x}}^{1}) + \sqrt[3]{x} \cdot 25 + 5 \sqrt[3]{x^{2}} + 5 (- 5 \sqrt[3]{x}) + 5 \cdot 25$

Step 2.4.3

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

$x - 5 {\sqrt[3]{x}}^{1 + 1} + \sqrt[3]{x} \cdot 25 + 5 \sqrt[3]{x^{2}} + 5 (- 5 \sqrt[3]{x}) + 5 \cdot 25$

Step 2.4.4

Add $1$ and $1$ .

$x - 5 {\sqrt[3]{x}}^{2} + \sqrt[3]{x} \cdot 25 + 5 \sqrt[3]{x^{2}} + 5 (- 5 \sqrt[3]{x}) + 5 \cdot 25$

Step 2.5

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

$x - 5 \sqrt[3]{x^{2}} + \sqrt[3]{x} \cdot 25 + 5 \sqrt[3]{x^{2}} + 5 (- 5 \sqrt[3]{x}) + 5 \cdot 25$

Step 2.6

Move $25$ to the left of $\sqrt[3]{x}$ .

$x - 5 \sqrt[3]{x^{2}} + 25 \cdot \sqrt[3]{x} + 5 \sqrt[3]{x^{2}} + 5 (- 5 \sqrt[3]{x}) + 5 \cdot 25$

Step 2.7

Multiply $- 5$ by $5$ .

$x - 5 \sqrt[3]{x^{2}} + 25 \sqrt[3]{x} + 5 \sqrt[3]{x^{2}} - 25 \sqrt[3]{x} + 5 \cdot 25$

Step 2.8

Multiply $5$ by $25$ .

$x - 5 \sqrt[3]{x^{2}} + 25 \sqrt[3]{x} + 5 \sqrt[3]{x^{2}} - 25 \sqrt[3]{x} + 125$

Step 3

Combine the opposite terms in $x - 5 \sqrt[3]{x^{2}} + 25 \sqrt[3]{x} + 5 \sqrt[3]{x^{2}} - 25 \sqrt[3]{x} + 125$ .

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Step 3.1

Add $- 5 \sqrt[3]{x^{2}}$ and $5 \sqrt[3]{x^{2}}$ .

$x + 0 + 25 \sqrt[3]{x} - 25 \sqrt[3]{x} + 125$

Step 3.2

Add $x$ and $0$ .

$x + 25 \sqrt[3]{x} - 25 \sqrt[3]{x} + 125$

Step 3.3

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

$x + 0 + 125$

Step 3.4

Add $x$ and $0$ .

$x + 125$