Chemistry Examples

$F_{1} \cdot \cos (45) + F_{2} \cdot \cos (30) = 500$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{F_{1} \sqrt{2}}{2} + F_{2} \cdot \cos (30) = 500$

Step 1.1.2

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{F_{1} \sqrt{2}}{2} + \frac{F_{2} \sqrt{3}}{2} = 500$

Step 2

Subtract $\frac{F_{2} \sqrt{3}}{2}$ from both sides of the equation.

$\frac{F_{1} \sqrt{2}}{2} = 500 - \frac{F_{2} \sqrt{3}}{2}$

Step 3

Multiply both sides of the equation by $\frac{2}{\sqrt{2}}$ .

$\frac{2}{\sqrt{2}} \cdot \frac{F_{1} \sqrt{2}}{2} = \frac{2}{\sqrt{2}} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{2}{\sqrt{2}} \cdot \frac{F_{1} \sqrt{2}}{2}$ .

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Step 4.1.1.1

Cancel the common factor of $2$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{2}{\sqrt{2}} \cdot \frac{F_{1} \sqrt{2}}{2} = \frac{2}{\sqrt{2}} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{\sqrt{2}} (F_{1} \sqrt{2}) = \frac{2}{\sqrt{2}} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.1.1.2

Cancel the common factor of $\sqrt{2}$ .

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Step 4.1.1.2.1

Factor $\sqrt{2}$ out of $F_{1} \sqrt{2}$ .

$\frac{1}{\sqrt{2}} (\sqrt{2} F_{1}) = \frac{2}{\sqrt{2}} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{1}{\sqrt{2}} (\sqrt{2} F_{1}) = \frac{2}{\sqrt{2}} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.1.1.2.3

Rewrite the expression.

$F_{1} = \frac{2}{\sqrt{2}} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{2}{\sqrt{2}} (500 - \frac{F_{2} \sqrt{3}}{2})$ .

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Step 4.2.1.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$F_{1} = \frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.2.1.2

Combine and simplify the denominator.

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Step 4.2.1.2.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$F_{1} = \frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.2.1.2.2

Raise $\sqrt{2}$ to the power of $1$ .

$F_{1} = \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.2.1.2.3

Raise $\sqrt{2}$ to the power of $1$ .

$F_{1} = \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.2.1.2.4

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

$F_{1} = \frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.2.1.2.5

Add $1$ and $1$ .

$F_{1} = \frac{2 \sqrt{2}}{{\sqrt{2}}^{2}} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.2.1.2.6

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

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Step 4.2.1.2.6.1

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

$F_{1} = \frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.2.1.2.6.2

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

$F_{1} = \frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.2.1.2.6.3

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

$F_{1} = \frac{2 \sqrt{2}}{2^{\frac{2}{2}}} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 4.2.1.2.6.4.1

Cancel the common factor.

$F_{1} = \frac{2 \sqrt{2}}{2^{\frac{2}{2}}} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.2.1.2.6.4.2

Rewrite the expression.

$F_{1} = \frac{2 \sqrt{2}}{2^{1}} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.2.1.2.6.5

Evaluate the exponent.

$F_{1} = \frac{2 \sqrt{2}}{2} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.2.1.3

Cancel the common factor of $2$ .

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Step 4.2.1.3.1

Cancel the common factor.

$F_{1} = \frac{2 \sqrt{2}}{2} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.2.1.3.2

Divide $\sqrt{2}$ by $1$ .

$F_{1} = \sqrt{2} (500 - \frac{F_{2} \sqrt{3}}{2})$

Step 4.2.1.4

Apply the distributive property.

$F_{1} = \sqrt{2} \cdot 500 + \sqrt{2} (- \frac{F_{2} \sqrt{3}}{2})$

Step 4.2.1.5

Move $500$ to the left of $\sqrt{2}$ .

$F_{1} = 500 \cdot \sqrt{2} + \sqrt{2} (- \frac{F_{2} \sqrt{3}}{2})$

Step 4.2.1.6

Multiply $\sqrt{2} (- \frac{F_{2} \sqrt{3}}{2})$ .

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Step 4.2.1.6.1

Combine $\sqrt{2}$ and $\frac{F_{2} \sqrt{3}}{2}$ .

$F_{1} = 500 \cdot \sqrt{2} - \frac{\sqrt{2} (F_{2} \sqrt{3})}{2}$

Step 4.2.1.6.2

Combine using the product rule for radicals.

$F_{1} = 500 \cdot \sqrt{2} - \frac{F_{2} \sqrt{2 \cdot 3}}{2}$

Step 4.2.1.6.3

Multiply $2$ by $3$ .

$F_{1} = 500 \cdot \sqrt{2} - \frac{F_{2} \sqrt{6}}{2}$

$F_{1} = 500 \sqrt{2} - \frac{F_{2} \sqrt{6}}{2}$

Step 5

Reorder $500 \sqrt{2}$ and $- \frac{F_{2} \sqrt{6}}{2}$ .

$F_{1} = - \frac{F_{2} \sqrt{6}}{2} + 500 \sqrt{2}$