Pre-Algebra Examples

$(\sqrt{87} - \sqrt{67}) (\sqrt{87} + \sqrt{67})$

Step 1

Expand $(\sqrt{87} - \sqrt{67}) (\sqrt{87} + \sqrt{67})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{87} (\sqrt{87} + \sqrt{67}) - \sqrt{67} (\sqrt{87} + \sqrt{67})$

Step 1.2

Apply the distributive property.

$\sqrt{87} \sqrt{87} + \sqrt{87} \sqrt{67} - \sqrt{67} (\sqrt{87} + \sqrt{67})$

Step 1.3

Apply the distributive property.

$\sqrt{87} \sqrt{87} + \sqrt{87} \sqrt{67} - \sqrt{67} \sqrt{87} - \sqrt{67} \sqrt{67}$

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.

$\sqrt{87 \cdot 87} + \sqrt{87} \sqrt{67} - \sqrt{67} \sqrt{87} - \sqrt{67} \sqrt{67}$

Step 2.1.2

Multiply $87$ by $87$ .

$\sqrt{7569} + \sqrt{87} \sqrt{67} - \sqrt{67} \sqrt{87} - \sqrt{67} \sqrt{67}$

Step 2.1.3

Rewrite $7569$ as $87^{2}$ .

$\sqrt{87^{2}} + \sqrt{87} \sqrt{67} - \sqrt{67} \sqrt{87} - \sqrt{67} \sqrt{67}$

Step 2.1.4

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

$87 + \sqrt{87} \sqrt{67} - \sqrt{67} \sqrt{87} - \sqrt{67} \sqrt{67}$

Step 2.1.5

Combine using the product rule for radicals.

$87 + \sqrt{87 \cdot 67} - \sqrt{67} \sqrt{87} - \sqrt{67} \sqrt{67}$

Step 2.1.6

Multiply $87$ by $67$ .

$87 + \sqrt{5829} - \sqrt{67} \sqrt{87} - \sqrt{67} \sqrt{67}$

Step 2.1.7

Multiply $- \sqrt{67} \sqrt{87}$ .

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Step 2.1.7.1

Combine using the product rule for radicals.

$87 + \sqrt{5829} - \sqrt{87 \cdot 67} - \sqrt{67} \sqrt{67}$

Step 2.1.7.2

Multiply $87$ by $67$ .

$87 + \sqrt{5829} - \sqrt{5829} - \sqrt{67} \sqrt{67}$

Step 2.1.8

Multiply $- \sqrt{67} \sqrt{67}$ .

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Step 2.1.8.1

Raise $\sqrt{67}$ to the power of $1$ .

$87 + \sqrt{5829} - \sqrt{5829} - ({\sqrt{67}}^{1} \sqrt{67})$

Step 2.1.8.2

Raise $\sqrt{67}$ to the power of $1$ .

$87 + \sqrt{5829} - \sqrt{5829} - ({\sqrt{67}}^{1} {\sqrt{67}}^{1})$

Step 2.1.8.3

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

$87 + \sqrt{5829} - \sqrt{5829} - {\sqrt{67}}^{1 + 1}$

Step 2.1.8.4

Add $1$ and $1$ .

$87 + \sqrt{5829} - \sqrt{5829} - {\sqrt{67}}^{2}$

Step 2.1.9

Rewrite ${\sqrt{67}}^{2}$ as $67$ .

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Step 2.1.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{67}$ as $67^{\frac{1}{2}}$ .

$87 + \sqrt{5829} - \sqrt{5829} - {(67^{\frac{1}{2}})}^{2}$

Step 2.1.9.2

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

$87 + \sqrt{5829} - \sqrt{5829} - 67^{\frac{1}{2} \cdot 2}$

Step 2.1.9.3

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

$87 + \sqrt{5829} - \sqrt{5829} - 67^{\frac{2}{2}}$

Step 2.1.9.4

Cancel the common factor of $2$ .

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Step 2.1.9.4.1

Cancel the common factor.

$87 + \sqrt{5829} - \sqrt{5829} - 67^{\frac{2}{2}}$

Step 2.1.9.4.2

Rewrite the expression.

$87 + \sqrt{5829} - \sqrt{5829} - 67^{1}$

Step 2.1.9.5

Evaluate the exponent.

$87 + \sqrt{5829} - \sqrt{5829} - 1 \cdot 67$

Step 2.1.10

Multiply $- 1$ by $67$ .

$87 + \sqrt{5829} - \sqrt{5829} - 67$

Step 2.2

Subtract $67$ from $87$ .

$20 + \sqrt{5829} - \sqrt{5829}$

Step 2.3

Subtract $\sqrt{5829}$ from $\sqrt{5829}$ .

$20 + 0$

Step 2.4

Add $20$ and $0$ .

$20$