Algebra Examples

$R = \sqrt{{(F x)}^{2} + {(F y)}^{2}}$

Step 1

Rewrite the equation as $\sqrt{{(F x)}^{2} + {(F y)}^{2}} = R$ .

$\sqrt{{(F x)}^{2} + {(F y)}^{2}} = R$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{{(F x)}^{2} + {(F y)}^{2}}}^{2} = R^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

${({({(F x)}^{2} + {(F y)}^{2})}^{\frac{1}{2}})}^{2} = R^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({({(F x)}^{2} + {(F y)}^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({({(F x)}^{2} + {(F y)}^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${({(F x)}^{2} + {(F y)}^{2})}^{\frac{1}{2} \cdot 2} = R^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${({(F x)}^{2} + {(F y)}^{2})}^{\frac{1}{2} \cdot 2} = R^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${({(F x)}^{2} + {(F y)}^{2})}^{1} = R^{2}$

Step 3.2.1.2

Simplify each term.

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Step 3.2.1.2.1

Apply the product rule to $F x$ .

${(F^{2} x^{2} + {(F y)}^{2})}^{1} = R^{2}$

Step 3.2.1.2.2

Apply the product rule to $F y$ .

${(F^{2} x^{2} + F^{2} y^{2})}^{1} = R^{2}$

Step 3.2.1.3

Simplify.

$F^{2} x^{2} + F^{2} y^{2} = R^{2}$

Step 4

Solve for $F$ .

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Step 4.1

Factor $F^{2}$ out of $F^{2} x^{2} + F^{2} y^{2}$ .

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Step 4.1.1

Factor $F^{2}$ out of $F^{2} x^{2}$ .

$F^{2} (x^{2}) + F^{2} y^{2} = R^{2}$

Step 4.1.2

Factor $F^{2}$ out of $F^{2} y^{2}$ .

$F^{2} (x^{2}) + F^{2} (y^{2}) = R^{2}$

Step 4.1.3

Factor $F^{2}$ out of $F^{2} (x^{2}) + F^{2} (y^{2})$ .

$F^{2} (x^{2} + y^{2}) = R^{2}$

Step 4.2

Divide each term in $F^{2} (x^{2} + y^{2}) = R^{2}$ by $x^{2} + y^{2}$ and simplify.

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Step 4.2.1

Divide each term in $F^{2} (x^{2} + y^{2}) = R^{2}$ by $x^{2} + y^{2}$ .

$\frac{F^{2} (x^{2} + y^{2})}{x^{2} + y^{2}} = \frac{R^{2}}{x^{2} + y^{2}}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $x^{2} + y^{2}$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{F^{2} (x^{2} + y^{2})}{x^{2} + y^{2}} = \frac{R^{2}}{x^{2} + y^{2}}$

Step 4.2.2.1.2

Divide $F^{2}$ by $1$ .

$F^{2} = \frac{R^{2}}{x^{2} + y^{2}}$

Step 4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$F = \pm \sqrt{\frac{R^{2}}{x^{2} + y^{2}}}$

Step 4.4

Simplify $\pm \sqrt{\frac{R^{2}}{x^{2} + y^{2}}}$ .

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Step 4.4.1

Rewrite $\sqrt{\frac{R^{2}}{x^{2} + y^{2}}}$ as $\frac{\sqrt{R^{2}}}{\sqrt{x^{2} + y^{2}}}$ .

$F = \pm \frac{\sqrt{R^{2}}}{\sqrt{x^{2} + y^{2}}}$

Step 4.4.2

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

$F = \pm \frac{R}{\sqrt{x^{2} + y^{2}}}$

Step 4.4.3

Multiply $\frac{R}{\sqrt{x^{2} + y^{2}}}$ by $\frac{\sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}}$ .

$F = \pm \frac{R}{\sqrt{x^{2} + y^{2}}} \cdot \frac{\sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}}$

Step 4.4.4

Combine and simplify the denominator.

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Step 4.4.4.1

Multiply $\frac{R}{\sqrt{x^{2} + y^{2}}}$ by $\frac{\sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}}$ .

$F = \pm \frac{R \sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}} \sqrt{x^{2} + y^{2}}}$

Step 4.4.4.2

Raise $\sqrt{x^{2} + y^{2}}$ to the power of $1$ .

$F = \pm \frac{R \sqrt{x^{2} + y^{2}}}{{\sqrt{x^{2} + y^{2}}}^{1} \sqrt{x^{2} + y^{2}}}$

Step 4.4.4.3

Raise $\sqrt{x^{2} + y^{2}}$ to the power of $1$ .

$F = \pm \frac{R \sqrt{x^{2} + y^{2}}}{{\sqrt{x^{2} + y^{2}}}^{1} {\sqrt{x^{2} + y^{2}}}^{1}}$

Step 4.4.4.4

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

$F = \pm \frac{R \sqrt{x^{2} + y^{2}}}{{\sqrt{x^{2} + y^{2}}}^{1 + 1}}$

Step 4.4.4.5

Add $1$ and $1$ .

$F = \pm \frac{R \sqrt{x^{2} + y^{2}}}{{\sqrt{x^{2} + y^{2}}}^{2}}$

Step 4.4.4.6

Rewrite ${\sqrt{x^{2} + y^{2}}}^{2}$ as $x^{2} + y^{2}$ .

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Step 4.4.4.6.1

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

$F = \pm \frac{R \sqrt{x^{2} + y^{2}}}{{({(x^{2} + y^{2})}^{\frac{1}{2}})}^{2}}$

Step 4.4.4.6.2

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

$F = \pm \frac{R \sqrt{x^{2} + y^{2}}}{{(x^{2} + y^{2})}^{\frac{1}{2} \cdot 2}}$

Step 4.4.4.6.3

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

$F = \pm \frac{R \sqrt{x^{2} + y^{2}}}{{(x^{2} + y^{2})}^{\frac{2}{2}}}$

Step 4.4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.4.6.4.1

Cancel the common factor.

$F = \pm \frac{R \sqrt{x^{2} + y^{2}}}{{(x^{2} + y^{2})}^{\frac{2}{2}}}$

Step 4.4.4.6.4.2

Rewrite the expression.

$F = \pm \frac{R \sqrt{x^{2} + y^{2}}}{{(x^{2} + y^{2})}^{1}}$

Step 4.4.4.6.5

Simplify.

$F = \pm \frac{R \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2}}$

Step 4.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.5.1

First, use the positive value of the $\pm$ to find the first solution.

$F = \frac{R \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2}}$

Step 4.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$F = - \frac{R \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2}}$

Step 4.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$F = \frac{R \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2}}$

$F = - \frac{R \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2}}$

$F = \frac{R \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2}}$

$F = - \frac{R \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2}}$

$F = \frac{R \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2}}$

$F = - \frac{R \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2}}$