Solve for r q=9r^2+16s^2r^2

$q = 9 r^{2} + 16 s^{2} r^{2}$

Step 1

Rewrite the equation as $9 r^{2} + 16 s^{2} r^{2} = q$ .

$9 r^{2} + 16 s^{2} r^{2} = q$

Step 2

Factor $r^{2}$ out of $9 r^{2} + 16 s^{2} r^{2}$ .

Factor $r^{2}$ out of $9 r^{2}$ .

$r^{2} \cdot 9 + 16 s^{2} r^{2} = q$

Factor $r^{2}$ out of $16 s^{2} r^{2}$ .

$r^{2} \cdot 9 + r^{2} (16 s^{2}) = q$

Factor $r^{2}$ out of $r^{2} \cdot 9 + r^{2} (16 s^{2})$ .

$r^{2} (9 + 16 s^{2}) = q$

Step 3

Divide each term in $r^{2} (9 + 16 s^{2}) = q$ by $9 + 16 s^{2}$ and simplify.

Divide each term in $r^{2} (9 + 16 s^{2}) = q$ by $9 + 16 s^{2}$ .

$\frac{r^{2} (9 + 16 s^{2})}{9 + 16 s^{2}} = \frac{q}{9 + 16 s^{2}}$

Simplify the left side.

Cancel the common factor of $9 + 16 s^{2}$ .

Cancel the common factor.

$\frac{r^{2} (9 + 16 s^{2})}{9 + 16 s^{2}} = \frac{q}{9 + 16 s^{2}}$

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

$r^{2} = \frac{q}{9 + 16 s^{2}}$

Step 4

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

$r = \pm \sqrt{\frac{q}{9 + 16 s^{2}}}$

Step 5

Simplify $\pm \sqrt{\frac{q}{9 + 16 s^{2}}}$ .

Rewrite $\sqrt{\frac{q}{9 + 16 s^{2}}}$ as $\frac{\sqrt{q}}{\sqrt{9 + 16 s^{2}}}$ .

$r = \pm \frac{\sqrt{q}}{\sqrt{9 + 16 s^{2}}}$

Multiply $\frac{\sqrt{q}}{\sqrt{9 + 16 s^{2}}}$ by $\frac{\sqrt{9 + 16 s^{2}}}{\sqrt{9 + 16 s^{2}}}$ .

$r = \pm \frac{\sqrt{q}}{\sqrt{9 + 16 s^{2}}} \cdot \frac{\sqrt{9 + 16 s^{2}}}{\sqrt{9 + 16 s^{2}}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{q}}{\sqrt{9 + 16 s^{2}}}$ by $\frac{\sqrt{9 + 16 s^{2}}}{\sqrt{9 + 16 s^{2}}}$ .

$r = \pm \frac{\sqrt{q} \sqrt{9 + 16 s^{2}}}{\sqrt{9 + 16 s^{2}} \sqrt{9 + 16 s^{2}}}$

Raise $\sqrt{9 + 16 s^{2}}$ to the power of $1$ .

$r = \pm \frac{\sqrt{q} \sqrt{9 + 16 s^{2}}}{{\sqrt{9 + 16 s^{2}}}^{1} \sqrt{9 + 16 s^{2}}}$

Raise $\sqrt{9 + 16 s^{2}}$ to the power of $1$ .

$r = \pm \frac{\sqrt{q} \sqrt{9 + 16 s^{2}}}{{\sqrt{9 + 16 s^{2}}}^{1} {\sqrt{9 + 16 s^{2}}}^{1}}$

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

$r = \pm \frac{\sqrt{q} \sqrt{9 + 16 s^{2}}}{{\sqrt{9 + 16 s^{2}}}^{1 + 1}}$

Add $1$ and $1$ .

$r = \pm \frac{\sqrt{q} \sqrt{9 + 16 s^{2}}}{{\sqrt{9 + 16 s^{2}}}^{2}}$

Rewrite ${\sqrt{9 + 16 s^{2}}}^{2}$ as $9 + 16 s^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 + 16 s^{2}}$ as ${(9 + 16 s^{2})}^{\frac{1}{2}}$ .

$r = \pm \frac{\sqrt{q} \sqrt{9 + 16 s^{2}}}{{({(9 + 16 s^{2})}^{\frac{1}{2}})}^{2}}$

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

$r = \pm \frac{\sqrt{q} \sqrt{9 + 16 s^{2}}}{{(9 + 16 s^{2})}^{\frac{1}{2} \cdot 2}}$

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

$r = \pm \frac{\sqrt{q} \sqrt{9 + 16 s^{2}}}{{(9 + 16 s^{2})}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$r = \pm \frac{\sqrt{q} \sqrt{9 + 16 s^{2}}}{{(9 + 16 s^{2})}^{\frac{2}{2}}}$

Rewrite the expression.

$r = \pm \frac{\sqrt{q} \sqrt{9 + 16 s^{2}}}{{(9 + 16 s^{2})}^{1}}$

Simplify.

$r = \pm \frac{\sqrt{q} \sqrt{9 + 16 s^{2}}}{9 + 16 s^{2}}$

Combine using the product rule for radicals.

$r = \pm \frac{\sqrt{q (9 + 16 s^{2})}}{9 + 16 s^{2}}$

Step 6

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.

$r = \frac{\sqrt{q (9 + 16 s^{2})}}{9 + 16 s^{2}}$

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

$r = - \frac{\sqrt{q (9 + 16 s^{2})}}{9 + 16 s^{2}}$

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

$r = \frac{\sqrt{q (9 + 16 s^{2})}}{9 + 16 s^{2}}$

$r = - \frac{\sqrt{q (9 + 16 s^{2})}}{9 + 16 s^{2}}$

$r = \frac{\sqrt{q (9 + 16 s^{2})}}{9 + 16 s^{2}}$

$r = - \frac{\sqrt{q (9 + 16 s^{2})}}{9 + 16 s^{2}}$

Solve for r q=9r^2+16s^2r^2

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