Solve for r f/(r^2)+a=n+k

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

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.

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

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

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.

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

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

Simplify the left side.

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Solve the equation.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Simplify the left side.

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

Cancel the common factor.

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

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

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

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

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

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

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

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Combine using the product rule for radicals.

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

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

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

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

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

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

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

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

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

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

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

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

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

Solve for r f/(r^2)+a=n+k

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