Algebra Examples

${(\sqrt{c} - \sqrt{d})}^{2}$

Step 1

Rewrite ${(\sqrt{c} - \sqrt{d})}^{2}$ as $(\sqrt{c} - \sqrt{d}) (\sqrt{c} - \sqrt{d})$ .

$(\sqrt{c} - \sqrt{d}) (\sqrt{c} - \sqrt{d})$

Step 2

Expand $(\sqrt{c} - \sqrt{d}) (\sqrt{c} - \sqrt{d})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\sqrt{c} (\sqrt{c} - \sqrt{d}) - \sqrt{d} (\sqrt{c} - \sqrt{d})$

Step 2.2

Apply the distributive property.

$\sqrt{c} \sqrt{c} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} (\sqrt{c} - \sqrt{d})$

Step 2.3

Apply the distributive property.

$\sqrt{c} \sqrt{c} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $\sqrt{c} \sqrt{c}$ .

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Step 3.1.1.1

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

${\sqrt{c}}^{1} \sqrt{c} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})$

Step 3.1.1.2

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

${\sqrt{c}}^{1} {\sqrt{c}}^{1} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})$

Step 3.1.1.3

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

${\sqrt{c}}^{1 + 1} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})$

Step 3.1.1.4

Add $1$ and $1$ .

${\sqrt{c}}^{2} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})$

Step 3.1.2

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

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Step 3.1.2.1

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

${(c^{\frac{1}{2}})}^{2} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})$

Step 3.1.2.2

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

$c^{\frac{1}{2} \cdot 2} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})$

Step 3.1.2.3

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

$c^{\frac{2}{2}} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})$

Step 3.1.2.4

Cancel the common factor of $2$ .

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Step 3.1.2.4.1

Cancel the common factor.

$c^{\frac{2}{2}} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})$

Step 3.1.2.4.2

Rewrite the expression.

$c^{1} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})$

Step 3.1.2.5

Simplify.

$c + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})$

Step 3.1.3

Rewrite using the commutative property of multiplication.

$c - \sqrt{c} \sqrt{d} - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})$

Step 3.1.4

Combine using the product rule for radicals.

$c - \sqrt{d c} - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})$

Step 3.1.5

Combine using the product rule for radicals.

$c - \sqrt{d c} - \sqrt{c d} - \sqrt{d} (- \sqrt{d})$

Step 3.1.6

Multiply $- \sqrt{d} (- \sqrt{d})$ .

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Step 3.1.6.1

Multiply $- 1$ by $- 1$ .

$c - \sqrt{d c} - \sqrt{c d} + 1 \sqrt{d} \sqrt{d}$

Step 3.1.6.2

Multiply $\sqrt{d}$ by $1$ .

$c - \sqrt{d c} - \sqrt{c d} + \sqrt{d} \sqrt{d}$

Step 3.1.6.3

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

$c - \sqrt{d c} - \sqrt{c d} + {\sqrt{d}}^{1} \sqrt{d}$

Step 3.1.6.4

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

$c - \sqrt{d c} - \sqrt{c d} + {\sqrt{d}}^{1} {\sqrt{d}}^{1}$

Step 3.1.6.5

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

$c - \sqrt{d c} - \sqrt{c d} + {\sqrt{d}}^{1 + 1}$

Step 3.1.6.6

Add $1$ and $1$ .

$c - \sqrt{d c} - \sqrt{c d} + {\sqrt{d}}^{2}$

Step 3.1.7

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

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Step 3.1.7.1

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

$c - \sqrt{d c} - \sqrt{c d} + {(d^{\frac{1}{2}})}^{2}$

Step 3.1.7.2

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

$c - \sqrt{d c} - \sqrt{c d} + d^{\frac{1}{2} \cdot 2}$

Step 3.1.7.3

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

$c - \sqrt{d c} - \sqrt{c d} + d^{\frac{2}{2}}$

Step 3.1.7.4

Cancel the common factor of $2$ .

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Step 3.1.7.4.1

Cancel the common factor.

$c - \sqrt{d c} - \sqrt{c d} + d^{\frac{2}{2}}$

Step 3.1.7.4.2

Rewrite the expression.

$c - \sqrt{d c} - \sqrt{c d} + d^{1}$

Step 3.1.7.5

Simplify.

$c - \sqrt{d c} - \sqrt{c d} + d$

Step 3.2

Subtract $\sqrt{c d}$ from $- \sqrt{d c}$ .

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Step 3.2.1

Reorder $d$ and $c$ .

$c - \sqrt{c d} - \sqrt{c d} + d$

Step 3.2.2

Subtract $\sqrt{c d}$ from $- \sqrt{c d}$ .

$c - 2 \sqrt{c d} + d$

Step 4

Move $- 2 \sqrt{c d}$ .

$c + d - 2 \sqrt{c d}$