Solve for m 4m^2n=49

$4 m^{2} n = 49$

Step 1

Divide each term in $4 m^{2} n = 49$ by $4 n$ and simplify.

Divide each term in $4 m^{2} n = 49$ by $4 n$ .

$\frac{4 m^{2} n}{4 n} = \frac{49}{4 n}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 m^{2} n}{4 n} = \frac{49}{4 n}$

Rewrite the expression.

$\frac{m^{2} n}{n} = \frac{49}{4 n}$

Cancel the common factor of $n$ .

Cancel the common factor.

$\frac{m^{2} n}{n} = \frac{49}{4 n}$

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

$m^{2} = \frac{49}{4 n}$

Step 2

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

$m = \pm \sqrt{\frac{49}{4 n}}$

Step 3

Simplify $\pm \sqrt{\frac{49}{4 n}}$ .

Rewrite $\frac{49}{4 n}$ as ${(\frac{7}{2})}^{2} \frac{1}{n}$ .

Factor the perfect power $7^{2}$ out of $49$ .

$m = \pm \sqrt{\frac{7^{2} \cdot 1}{4 n}}$

Factor the perfect power $2^{2}$ out of $4 n$ .

$m = \pm \sqrt{\frac{7^{2} \cdot 1}{2^{2} n}}$

Rearrange the fraction $\frac{7^{2} \cdot 1}{2^{2} n}$ .

$m = \pm \sqrt{{(\frac{7}{2})}^{2} \frac{1}{n}}$

Pull terms out from under the radical.

$m = \pm \frac{7}{2} \sqrt{\frac{1}{n}}$

Rewrite $\sqrt{\frac{1}{n}}$ as $\frac{\sqrt{1}}{\sqrt{n}}$ .

$m = \pm \frac{7}{2} \cdot \frac{\sqrt{1}}{\sqrt{n}}$

Any root of $1$ is $1$ .

$m = \pm \frac{7}{2} \cdot \frac{1}{\sqrt{n}}$

Multiply $\frac{1}{\sqrt{n}}$ by $\frac{\sqrt{n}}{\sqrt{n}}$ .

$m = \pm \frac{7}{2} (\frac{1}{\sqrt{n}} \cdot \frac{\sqrt{n}}{\sqrt{n}})$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{n}}$ by $\frac{\sqrt{n}}{\sqrt{n}}$ .

$m = \pm \frac{7}{2} \cdot \frac{\sqrt{n}}{\sqrt{n} \sqrt{n}}$

Raise $\sqrt{n}$ to the power of $1$ .

$m = \pm \frac{7}{2} \cdot \frac{\sqrt{n}}{{\sqrt{n}}^{1} \sqrt{n}}$

Raise $\sqrt{n}$ to the power of $1$ .

$m = \pm \frac{7}{2} \cdot \frac{\sqrt{n}}{{\sqrt{n}}^{1} {\sqrt{n}}^{1}}$

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

$m = \pm \frac{7}{2} \cdot \frac{\sqrt{n}}{{\sqrt{n}}^{1 + 1}}$

Add $1$ and $1$ .

$m = \pm \frac{7}{2} \cdot \frac{\sqrt{n}}{{\sqrt{n}}^{2}}$

Rewrite ${\sqrt{n}}^{2}$ as $n$ .

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

$m = \pm \frac{7}{2} \cdot \frac{\sqrt{n}}{{(n^{\frac{1}{2}})}^{2}}$

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

$m = \pm \frac{7}{2} \cdot \frac{\sqrt{n}}{n^{\frac{1}{2} \cdot 2}}$

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

$m = \pm \frac{7}{2} \cdot \frac{\sqrt{n}}{n^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$m = \pm \frac{7}{2} \cdot \frac{\sqrt{n}}{n^{\frac{2}{2}}}$

Rewrite the expression.

$m = \pm \frac{7}{2} \cdot \frac{\sqrt{n}}{n^{1}}$

Simplify.

$m = \pm \frac{7}{2} \cdot \frac{\sqrt{n}}{n}$

Multiply $\frac{7}{2}$ by $\frac{\sqrt{n}}{n}$ .

$m = \pm \frac{7 \sqrt{n}}{2 n}$

Step 4

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.

$m = \frac{7 \sqrt{n}}{2 n}$

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

$m = - \frac{7 \sqrt{n}}{2 n}$

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

$m = \frac{7 \sqrt{n}}{2 n}, - \frac{7 \sqrt{n}}{2 n}$

Solve for m 4m^2n=49

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