Solve for r M=pr^2hd

$M = p r^{2} h d$

Step 1

Rewrite the equation as $p r^{2} h d = M$ .

$p r^{2} h d = M$

Step 2

Divide each term in $p r^{2} h d = M$ by $p h d$ and simplify.

Divide each term in $p r^{2} h d = M$ by $p h d$ .

$\frac{p r^{2} h d}{p h d} = \frac{M}{p h d}$

Simplify the left side.

Cancel the common factor of $p$ .

Cancel the common factor.

$\frac{p r^{2} h d}{p h d} = \frac{M}{p h d}$

Rewrite the expression.

$\frac{r^{2} h d}{h d} = \frac{M}{p h d}$

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{r^{2} h d}{h d} = \frac{M}{p h d}$

Rewrite the expression.

$\frac{r^{2} d}{d} = \frac{M}{p h d}$

Cancel the common factor of $d$ .

Cancel the common factor.

$\frac{r^{2} d}{d} = \frac{M}{p h d}$

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

$r^{2} = \frac{M}{p h d}$

Step 3

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

$r = \pm \sqrt{\frac{M}{p h d}}$

Step 4

Simplify $\pm \sqrt{\frac{M}{p h d}}$ .

Rewrite $\sqrt{\frac{M}{p h d}}$ as $\frac{\sqrt{M}}{\sqrt{p h d}}$ .

$r = \pm \frac{\sqrt{M}}{\sqrt{p h d}}$

Multiply $\frac{\sqrt{M}}{\sqrt{p h d}}$ by $\frac{\sqrt{p h d}}{\sqrt{p h d}}$ .

$r = \pm \frac{\sqrt{M}}{\sqrt{p h d}} \cdot \frac{\sqrt{p h d}}{\sqrt{p h d}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{M}}{\sqrt{p h d}}$ by $\frac{\sqrt{p h d}}{\sqrt{p h d}}$ .

$r = \pm \frac{\sqrt{M} \sqrt{p h d}}{\sqrt{p h d} \sqrt{p h d}}$

Raise $\sqrt{p h d}$ to the power of $1$ .

$r = \pm \frac{\sqrt{M} \sqrt{p h d}}{{\sqrt{p h d}}^{1} \sqrt{p h d}}$

Raise $\sqrt{p h d}$ to the power of $1$ .

$r = \pm \frac{\sqrt{M} \sqrt{p h d}}{{\sqrt{p h d}}^{1} {\sqrt{p h d}}^{1}}$

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

$r = \pm \frac{\sqrt{M} \sqrt{p h d}}{{\sqrt{p h d}}^{1 + 1}}$

Add $1$ and $1$ .

$r = \pm \frac{\sqrt{M} \sqrt{p h d}}{{\sqrt{p h d}}^{2}}$

Rewrite ${\sqrt{p h d}}^{2}$ as $p h d$ .

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

$r = \pm \frac{\sqrt{M} \sqrt{p h d}}{{({(p h d)}^{\frac{1}{2}})}^{2}}$

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

$r = \pm \frac{\sqrt{M} \sqrt{p h d}}{{(p h d)}^{\frac{1}{2} \cdot 2}}$

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

$r = \pm \frac{\sqrt{M} \sqrt{p h d}}{{(p h d)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$r = \pm \frac{\sqrt{M} \sqrt{p h d}}{{(p h d)}^{\frac{2}{2}}}$

Rewrite the expression.

$r = \pm \frac{\sqrt{M} \sqrt{p h d}}{{(p h d)}^{1}}$

Simplify.

$r = \pm \frac{\sqrt{M} \sqrt{p h d}}{p h d}$

Combine using the product rule for radicals.

$r = \pm \frac{\sqrt{M p h d}}{p h d}$

Step 5

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.

$r = \frac{\sqrt{M p h d}}{p h d}$

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

$r = - \frac{\sqrt{M p h d}}{p h d}$

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

$r = \frac{\sqrt{M p h d}}{p h d}$

$r = - \frac{\sqrt{M p h d}}{p h d}$

$r = \frac{\sqrt{M p h d}}{p h d}$

$r = - \frac{\sqrt{M p h d}}{p h d}$

Solve for r M=pr^2hd

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