Algebra Examples

$(\sqrt{3} + \sqrt{z}) (\sqrt{3} - \sqrt{z})$

Step 1

Expand $(\sqrt{3} + \sqrt{z}) (\sqrt{3} - \sqrt{z})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{3} (\sqrt{3} - \sqrt{z}) + \sqrt{z} (\sqrt{3} - \sqrt{z})$

Step 1.2

Apply the distributive property.

$\sqrt{3} \sqrt{3} + \sqrt{3} (- \sqrt{z}) + \sqrt{z} (\sqrt{3} - \sqrt{z})$

Step 1.3

Apply the distributive property.

$\sqrt{3} \sqrt{3} + \sqrt{3} (- \sqrt{z}) + \sqrt{z} \sqrt{3} + \sqrt{z} (- \sqrt{z})$

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $\sqrt{3} \sqrt{3} + \sqrt{3} (- \sqrt{z}) + \sqrt{z} \sqrt{3} + \sqrt{z} (- \sqrt{z})$ .

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Step 2.1.1

Reorder the factors in the terms $\sqrt{3} (- \sqrt{z})$ and $\sqrt{z} \sqrt{3}$ .

$\sqrt{3} \sqrt{3} - \sqrt{3} \sqrt{z} + \sqrt{3} \sqrt{z} + \sqrt{z} (- \sqrt{z})$

Step 2.1.2

Add $- \sqrt{3} \sqrt{z}$ and $\sqrt{3} \sqrt{z}$ .

$\sqrt{3} \sqrt{3} + 0 + \sqrt{z} (- \sqrt{z})$

Step 2.1.3

Add $\sqrt{3} \sqrt{3}$ and $0$ .

$\sqrt{3} \sqrt{3} + \sqrt{z} (- \sqrt{z})$

Step 2.2

Simplify each term.

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Step 2.2.1

Combine using the product rule for radicals.

$\sqrt{3 \cdot 3} + \sqrt{z} (- \sqrt{z})$

Step 2.2.2

Multiply $3$ by $3$ .

$\sqrt{9} + \sqrt{z} (- \sqrt{z})$

Step 2.2.3

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2}} + \sqrt{z} (- \sqrt{z})$

Step 2.2.4

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

$3 + \sqrt{z} (- \sqrt{z})$

Step 2.2.5

Rewrite using the commutative property of multiplication.

$3 - \sqrt{z} \sqrt{z}$

Step 2.2.6

Multiply $- \sqrt{z} \sqrt{z}$ .

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Step 2.2.6.1

Raise $\sqrt{z}$ to the power of $1$ .

$3 - ({\sqrt{z}}^{1} \sqrt{z})$

Step 2.2.6.2

Raise $\sqrt{z}$ to the power of $1$ .

$3 - ({\sqrt{z}}^{1} {\sqrt{z}}^{1})$

Step 2.2.6.3

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

$3 - {\sqrt{z}}^{1 + 1}$

Step 2.2.6.4

Add $1$ and $1$ .

$3 - {\sqrt{z}}^{2}$

Step 2.2.7

Rewrite ${\sqrt{z}}^{2}$ as $z$ .

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Step 2.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z}$ as $z^{\frac{1}{2}}$ .

$3 - {(z^{\frac{1}{2}})}^{2}$

Step 2.2.7.2

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

$3 - z^{\frac{1}{2} \cdot 2}$

Step 2.2.7.3

Combine $\frac{1}{2}$ and $2$ .

$3 - z^{\frac{2}{2}}$

Step 2.2.7.4

Cancel the common factor of $2$ .

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Step 2.2.7.4.1

Cancel the common factor.

$3 - z^{\frac{2}{2}}$

Step 2.2.7.4.2

Rewrite the expression.

$3 - z^{1}$

Step 2.2.7.5

Simplify.

$3 - z$