Algebra Examples

$\frac{f}{r^{2}} + a = n + k$

Step 1

Subtract $a$ from both sides of the equation.

$\frac{f}{r^{2}} = n + k - a$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$r^{2}, 1, 1, 1$

Step 2.2

The LCM of one and any expression is the expression.

$r^{2}$

Step 3

Multiply each term in $\frac{f}{r^{2}} = n + k - a$ by $r^{2}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{f}{r^{2}} = n + k - a$ by $r^{2}$ .

$\frac{f}{r^{2}} r^{2} = n r^{2} + k r^{2} - a r^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $r^{2}$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{f}{r^{2}} r^{2} = n r^{2} + k r^{2} - a r^{2}$

Step 3.2.1.2

Rewrite the expression.

$f = n r^{2} + k r^{2} - a r^{2}$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $n r^{2} + k r^{2} - a r^{2} = f$ .

$n r^{2} + k r^{2} - a r^{2} = f$

Step 4.2

Factor $r^{2}$ out of $n r^{2} + k r^{2} - a r^{2}$ .

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Step 4.2.1

Factor $r^{2}$ out of $n r^{2}$ .

$r^{2} n + k r^{2} - a r^{2} = f$

Step 4.2.2

Factor $r^{2}$ out of $k r^{2}$ .

$r^{2} n + r^{2} k - a r^{2} = f$

Step 4.2.3

Factor $r^{2}$ out of $- a r^{2}$ .

$r^{2} n + r^{2} k + r^{2} (- a) = f$

Step 4.2.4

Factor $r^{2}$ out of $r^{2} n + r^{2} k$ .

$r^{2} (n + k) + r^{2} (- a) = f$

Step 4.2.5

Factor $r^{2}$ out of $r^{2} (n + k) + r^{2} (- a)$ .

$r^{2} (n + k - a) = f$

Step 4.3

Divide each term in $r^{2} (n + k - a) = f$ by $n + k - a$ and simplify.

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Step 4.3.1

Divide each term in $r^{2} (n + k - a) = f$ by $n + k - a$ .

$\frac{r^{2} (n + k - a)}{n + k - a} = \frac{f}{n + k - a}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $n + k - a$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{r^{2} (n + k - a)}{n + k - a} = \frac{f}{n + k - a}$

Step 4.3.2.1.2

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

$r^{2} = \frac{f}{n + k - a}$

Step 4.4

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

$r = \pm \sqrt{\frac{f}{n + k - a}}$

Step 4.5

Simplify $\pm \sqrt{\frac{f}{n + k - a}}$ .

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Step 4.5.1

Rewrite $\sqrt{\frac{f}{n + k - a}}$ as $\frac{\sqrt{f}}{\sqrt{n + k - a}}$ .

$r = \pm \frac{\sqrt{f}}{\sqrt{n + k - a}}$

Step 4.5.2

Multiply $\frac{\sqrt{f}}{\sqrt{n + k - a}}$ by $\frac{\sqrt{n + k - a}}{\sqrt{n + k - a}}$ .

$r = \pm \frac{\sqrt{f}}{\sqrt{n + k - a}} \cdot \frac{\sqrt{n + k - a}}{\sqrt{n + k - a}}$

Step 4.5.3

Combine and simplify the denominator.

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Step 4.5.3.1

Multiply $\frac{\sqrt{f}}{\sqrt{n + k - a}}$ by $\frac{\sqrt{n + k - a}}{\sqrt{n + k - a}}$ .

$r = \pm \frac{\sqrt{f} \sqrt{n + k - a}}{\sqrt{n + k - a} \sqrt{n + k - a}}$

Step 4.5.3.2

Raise $\sqrt{n + k - a}$ to the power of $1$ .

$r = \pm \frac{\sqrt{f} \sqrt{n + k - a}}{{\sqrt{n + k - a}}^{1} \sqrt{n + k - a}}$

Step 4.5.3.3

Raise $\sqrt{n + k - a}$ to the power of $1$ .

$r = \pm \frac{\sqrt{f} \sqrt{n + k - a}}{{\sqrt{n + k - a}}^{1} {\sqrt{n + k - a}}^{1}}$

Step 4.5.3.4

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

$r = \pm \frac{\sqrt{f} \sqrt{n + k - a}}{{\sqrt{n + k - a}}^{1 + 1}}$

Step 4.5.3.5

Add $1$ and $1$ .

$r = \pm \frac{\sqrt{f} \sqrt{n + k - a}}{{\sqrt{n + k - a}}^{2}}$

Step 4.5.3.6

Rewrite ${\sqrt{n + k - a}}^{2}$ as $n + k - a$ .

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Step 4.5.3.6.1

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

$r = \pm \frac{\sqrt{f} \sqrt{n + k - a}}{{({(n + k - a)}^{\frac{1}{2}})}^{2}}$

Step 4.5.3.6.2

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

$r = \pm \frac{\sqrt{f} \sqrt{n + k - a}}{{(n + k - a)}^{\frac{1}{2} \cdot 2}}$

Step 4.5.3.6.3

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

$r = \pm \frac{\sqrt{f} \sqrt{n + k - a}}{{(n + k - a)}^{\frac{2}{2}}}$

Step 4.5.3.6.4

Cancel the common factor of $2$ .

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Step 4.5.3.6.4.1

Cancel the common factor.

$r = \pm \frac{\sqrt{f} \sqrt{n + k - a}}{{(n + k - a)}^{\frac{2}{2}}}$

Step 4.5.3.6.4.2

Rewrite the expression.

$r = \pm \frac{\sqrt{f} \sqrt{n + k - a}}{{(n + k - a)}^{1}}$

Step 4.5.3.6.5

Simplify.

$r = \pm \frac{\sqrt{f} \sqrt{n + k - a}}{n + k - a}$

Step 4.5.4

Combine using the product rule for radicals.

$r = \pm \frac{\sqrt{f (n + k - a)}}{n + k - a}$

Step 4.6

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

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Step 4.6.1

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

$r = \frac{\sqrt{f (n + k - a)}}{n + k - a}$

Step 4.6.2

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

$r = - \frac{\sqrt{f (n + k - a)}}{n + k - a}$

Step 4.6.3

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

$r = \frac{\sqrt{f (n + k - a)}}{n + k - a}$

$r = - \frac{\sqrt{f (n + k - a)}}{n + k - a}$

$r = \frac{\sqrt{f (n + k - a)}}{n + k - a}$

$r = - \frac{\sqrt{f (n + k - a)}}{n + k - a}$

$r = \frac{\sqrt{f (n + k - a)}}{n + k - a}$

$r = - \frac{\sqrt{f (n + k - a)}}{n + k - a}$