Trigonometry Examples

$m^{2} n + 3 {ps}^{3} = 2 q$

Step 1

Subtract $3 {ps}^{3}$ from both sides of the equation.

$m^{2} n = 2 q - (3 {ps}^{3})$

Step 2

Divide each term in $m^{2} n = 2 q - (3 {ps}^{3})$ by $n$ and simplify.

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Step 2.1

Divide each term in $m^{2} n = 2 q - (3 {ps}^{3})$ by $n$ .

$\frac{m^{2} n}{n} = \frac{2 q}{n} + \frac{- (3 {ps}^{3})}{n}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $n$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{m^{2} n}{n} = \frac{2 q}{n} + \frac{- (3 {ps}^{3})}{n}$

Step 2.2.1.2

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

$m^{2} = \frac{2 q}{n} + \frac{- (3 {ps}^{3})}{n}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$m^{2} = \frac{2 q}{n} - \frac{3 {ps}^{3}}{n}$

Step 3

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

$m = \pm \sqrt{\frac{2 q}{n} - \frac{3 {ps}^{3}}{n}}$

Step 4

Simplify $\pm \sqrt{\frac{2 q}{n} - \frac{3 {ps}^{3}}{n}}$ .

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Step 4.1

Combine the numerators over the common denominator.

$m = \pm \sqrt{\frac{2 q - (3 {ps}^{3})}{n}}$

Step 4.2

Rewrite $\sqrt{\frac{2 q - (3 {ps}^{3})}{n}}$ as $\frac{\sqrt{2 q - (3 {ps}^{3})}}{\sqrt{n}}$ .

$m = \pm \frac{\sqrt{2 q - (3 {ps}^{3})}}{\sqrt{n}}$

Step 4.3

Multiply $\frac{\sqrt{2 q - (3 {ps}^{3})}}{\sqrt{n}}$ by $\frac{\sqrt{n}}{\sqrt{n}}$ .

$m = \pm \frac{\sqrt{2 q - (3 {ps}^{3})}}{\sqrt{n}} \cdot \frac{\sqrt{n}}{\sqrt{n}}$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{\sqrt{2 q - (3 {ps}^{3})}}{\sqrt{n}}$ by $\frac{\sqrt{n}}{\sqrt{n}}$ .

$m = \pm \frac{\sqrt{2 q - (3 {ps}^{3})} \sqrt{n}}{\sqrt{n} \sqrt{n}}$

Step 4.4.2

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

$m = \pm \frac{\sqrt{2 q - (3 {ps}^{3})} \sqrt{n}}{{\sqrt{n}}^{1} \sqrt{n}}$

Step 4.4.3

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

$m = \pm \frac{\sqrt{2 q - (3 {ps}^{3})} \sqrt{n}}{{\sqrt{n}}^{1} {\sqrt{n}}^{1}}$

Step 4.4.4

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

$m = \pm \frac{\sqrt{2 q - (3 {ps}^{3})} \sqrt{n}}{{\sqrt{n}}^{1 + 1}}$

Step 4.4.5

Add $1$ and $1$ .

$m = \pm \frac{\sqrt{2 q - (3 {ps}^{3})} \sqrt{n}}{{\sqrt{n}}^{2}}$

Step 4.4.6

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

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Step 4.4.6.1

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

$m = \pm \frac{\sqrt{2 q - (3 {ps}^{3})} \sqrt{n}}{{(n^{\frac{1}{2}})}^{2}}$

Step 4.4.6.2

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

$m = \pm \frac{\sqrt{2 q - (3 {ps}^{3})} \sqrt{n}}{n^{\frac{1}{2} \cdot 2}}$

Step 4.4.6.3

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

$m = \pm \frac{\sqrt{2 q - (3 {ps}^{3})} \sqrt{n}}{n^{\frac{2}{2}}}$

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

$m = \pm \frac{\sqrt{2 q - (3 {ps}^{3})} \sqrt{n}}{n^{\frac{2}{2}}}$

Step 4.4.6.4.2

Rewrite the expression.

$m = \pm \frac{\sqrt{2 q - (3 {ps}^{3})} \sqrt{n}}{n^{1}}$

Step 4.4.6.5

Simplify.

$m = \pm \frac{\sqrt{2 q - (3 {ps}^{3})} \sqrt{n}}{n}$

Step 4.5

Combine using the product rule for radicals.

$m = \pm \frac{\sqrt{(2 q - (3 {ps}^{3})) n}}{n}$

Step 4.6

Reorder factors in $\pm \frac{\sqrt{(2 q - (3 {ps}^{3})) n}}{n}$ .

$m = \pm \frac{\sqrt{n (2 q - (3 {ps}^{3}))}}{n}$

Step 5

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

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Step 5.1

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

$m = \frac{\sqrt{n (2 q - (3 {ps}^{3}))}}{n}$

Step 5.2

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

$m = - \frac{\sqrt{n (2 q - (3 {ps}^{3}))}}{n}$

Step 5.3

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

$m = \frac{\sqrt{n (2 q - (3 {ps}^{3}))}}{n}, - \frac{\sqrt{n (2 q - (3 {ps}^{3}))}}{n}$

Step 6

Simplify $m = \frac{\sqrt{n (2 q - (3 {ps}^{3}))}}{n}, - \frac{\sqrt{n (2 q - (3 {ps}^{3}))}}{n}$ .

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Step 6.1

Multiply $- 1$ by $3$ .

$m = \frac{\sqrt{n (2 q - 3 {ps}^{3})}}{n}, - \frac{\sqrt{n (2 q - (3 {ps}^{3}))}}{n}$

Step 6.2

Multiply $- 1$ by $3$ .

$m = \frac{\sqrt{n (2 q - 3 {ps}^{3})}}{n}, - \frac{\sqrt{n (2 q - 3 {ps}^{3})}}{n}$