Basic Math Examples

(\sqrt{85} - \sqrt{65}) (\sqrt{85} + \sqrt{65})

$(\sqrt{85} - \sqrt{65}) (\sqrt{85} + \sqrt{65})$

Step 1

Expand

(\sqrt{85} - \sqrt{65}) (\sqrt{85} + \sqrt{65})

$(\sqrt{85} - \sqrt{65}) (\sqrt{85} + \sqrt{65})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

\sqrt{85} (\sqrt{85} + \sqrt{65}) - \sqrt{65} (\sqrt{85} + \sqrt{65})

$\sqrt{85} (\sqrt{85} + \sqrt{65}) - \sqrt{65} (\sqrt{85} + \sqrt{65})$

Step 1.2

Apply the distributive property.

\sqrt{85} \sqrt{85} + \sqrt{85} \sqrt{65} - \sqrt{65} (\sqrt{85} + \sqrt{65})

$\sqrt{85} \sqrt{85} + \sqrt{85} \sqrt{65} - \sqrt{65} (\sqrt{85} + \sqrt{65})$

Step 1.3

Apply the distributive property.

\sqrt{85} \sqrt{85} + \sqrt{85} \sqrt{65} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}

$\sqrt{85} \sqrt{85} + \sqrt{85} \sqrt{65} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}$

\sqrt{85} \sqrt{85} + \sqrt{85} \sqrt{65} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}

$\sqrt{85} \sqrt{85} + \sqrt{85} \sqrt{65} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine using the product rule for radicals.

\sqrt{85 \cdot 85} + \sqrt{85} \sqrt{65} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}

$\sqrt{85 \cdot 85} + \sqrt{85} \sqrt{65} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}$

Step 2.1.2

Multiply

85

$85$ by

85

$85$ .

\sqrt{7225} + \sqrt{85} \sqrt{65} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}

$\sqrt{7225} + \sqrt{85} \sqrt{65} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}$

Step 2.1.3

Rewrite

7225

$7225$ as

85^{2}

$85^{2}$ .

\sqrt{85^{2}} + \sqrt{85} \sqrt{65} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}

$\sqrt{85^{2}} + \sqrt{85} \sqrt{65} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}$

Step 2.1.4

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

85 + \sqrt{85} \sqrt{65} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}

$85 + \sqrt{85} \sqrt{65} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}$

Step 2.1.5

Combine using the product rule for radicals.

85 + \sqrt{85 \cdot 65} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}

$85 + \sqrt{85 \cdot 65} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}$

Step 2.1.6

Multiply

85

$85$ by

65

$65$ .

85 + \sqrt{5525} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}

$85 + \sqrt{5525} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}$

Step 2.1.7

Rewrite

5525

$5525$ as

5^{2} \cdot 221

$5^{2} \cdot 221$ .

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Step 2.1.7.1

Factor

25

$25$ out of

5525

$5525$ .

85 + \sqrt{25 (221)} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}

$85 + \sqrt{25 (221)} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}$

Step 2.1.7.2

Rewrite

25

$25$ as

5^{2}

$5^{2}$ .

85 + \sqrt{5^{2} \cdot 221} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}

$85 + \sqrt{5^{2} \cdot 221} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}$

85 + \sqrt{5^{2} \cdot 221} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}

$85 + \sqrt{5^{2} \cdot 221} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}$

Step 2.1.8

Pull terms out from under the radical.

85 + 5 \sqrt{221} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}

$85 + 5 \sqrt{221} - \sqrt{65} \sqrt{85} - \sqrt{65} \sqrt{65}$

Step 2.1.9

Multiply

- \sqrt{65} \sqrt{85}

$- \sqrt{65} \sqrt{85}$ .

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Step 2.1.9.1

Combine using the product rule for radicals.

85 + 5 \sqrt{221} - \sqrt{85 \cdot 65} - \sqrt{65} \sqrt{65}

$85 + 5 \sqrt{221} - \sqrt{85 \cdot 65} - \sqrt{65} \sqrt{65}$

Step 2.1.9.2

Multiply

85

$85$ by

65

$65$ .

85 + 5 \sqrt{221} - \sqrt{5525} - \sqrt{65} \sqrt{65}

$85 + 5 \sqrt{221} - \sqrt{5525} - \sqrt{65} \sqrt{65}$

85 + 5 \sqrt{221} - \sqrt{5525} - \sqrt{65} \sqrt{65}

$85 + 5 \sqrt{221} - \sqrt{5525} - \sqrt{65} \sqrt{65}$

Step 2.1.10

Rewrite

5525

$5525$ as

5^{2} \cdot 221

$5^{2} \cdot 221$ .

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Step 2.1.10.1

Factor

25

$25$ out of

5525

$5525$ .

85 + 5 \sqrt{221} - \sqrt{25 (221)} - \sqrt{65} \sqrt{65}

$85 + 5 \sqrt{221} - \sqrt{25 (221)} - \sqrt{65} \sqrt{65}$

Step 2.1.10.2

Rewrite

25

$25$ as

5^{2}

$5^{2}$ .

85 + 5 \sqrt{221} - \sqrt{5^{2} \cdot 221} - \sqrt{65} \sqrt{65}

$85 + 5 \sqrt{221} - \sqrt{5^{2} \cdot 221} - \sqrt{65} \sqrt{65}$

85 + 5 \sqrt{221} - \sqrt{5^{2} \cdot 221} - \sqrt{65} \sqrt{65}

$85 + 5 \sqrt{221} - \sqrt{5^{2} \cdot 221} - \sqrt{65} \sqrt{65}$

Step 2.1.11

Pull terms out from under the radical.

85 + 5 \sqrt{221} - (5 \sqrt{221}) - \sqrt{65} \sqrt{65}

$85 + 5 \sqrt{221} - (5 \sqrt{221}) - \sqrt{65} \sqrt{65}$

Step 2.1.12

Multiply

5

$5$ by

- 1

$- 1$ .

85 + 5 \sqrt{221} - 5 \sqrt{221} - \sqrt{65} \sqrt{65}

$85 + 5 \sqrt{221} - 5 \sqrt{221} - \sqrt{65} \sqrt{65}$

Step 2.1.13

Multiply

- \sqrt{65} \sqrt{65}

$- \sqrt{65} \sqrt{65}$ .

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Step 2.1.13.1

Raise

$\sqrt{65}$ to the power of

$1$ .

$85 + 5 \sqrt{221} - 5 \sqrt{221} - ({\sqrt{65}}^{1} \sqrt{65})$

Step 2.1.13.2

Raise

$\sqrt{65}$ to the power of

$1$ .

$85 + 5 \sqrt{221} - 5 \sqrt{221} - ({\sqrt{65}}^{1} {\sqrt{65}}^{1})$

Step 2.1.13.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$85 + 5 \sqrt{221} - 5 \sqrt{221} - {\sqrt{65}}^{1 + 1}$

Step 2.1.13.4

Add

$1$ and

$1$ .

$85 + 5 \sqrt{221} - 5 \sqrt{221} - {\sqrt{65}}^{2}$

Step 2.1.14

Rewrite

${\sqrt{65}}^{2}$ as

$65$ .

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Step 2.1.14.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{65}$ as

$65^{\frac{1}{2}}$ .

$85 + 5 \sqrt{221} - 5 \sqrt{221} - {(65^{\frac{1}{2}})}^{2}$

Step 2.1.14.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$85 + 5 \sqrt{221} - 5 \sqrt{221} - 65^{\frac{1}{2} \cdot 2}$

Step 2.1.14.3

Combine

$\frac{1}{2}$ and

$2$ .

$85 + 5 \sqrt{221} - 5 \sqrt{221} - 65^{\frac{2}{2}}$

Step 2.1.14.4

Cancel the common factor of

$2$ .

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Step 2.1.14.4.1

Cancel the common factor.

$85 + 5 \sqrt{221} - 5 \sqrt{221} - 65^{\frac{2}{2}}$

Step 2.1.14.4.2

Rewrite the expression.

$85 + 5 \sqrt{221} - 5 \sqrt{221} - 65^{1}$

Step 2.1.14.5

Evaluate the exponent.

$85 + 5 \sqrt{221} - 5 \sqrt{221} - 1 \cdot 65$

Step 2.1.15

Multiply

$- 1$ by

$65$ .

$85 + 5 \sqrt{221} - 5 \sqrt{221} - 65$

Step 2.2

Subtract

$65$ from

$85$ .

$20 + 5 \sqrt{221} - 5 \sqrt{221}$

Step 2.3

Subtract

$5 \sqrt{221}$ from

$5 \sqrt{221}$ .

$20 + 0$

Step 2.4

Add

$20$ and

$0$ .

$20$