Basic Math Examples

$5 c^{2} = - 2$

Step 1

Divide each term in $5 c^{2} = - 2$ by $5$ and simplify.

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Step 1.1

Divide each term in $5 c^{2} = - 2$ by $5$ .

$\frac{5 c^{2}}{5} = \frac{- 2}{5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{5 c^{2}}{5} = \frac{- 2}{5}$

Step 1.2.1.2

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

$c^{2} = \frac{- 2}{5}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$c^{2} = - \frac{2}{5}$

Step 2

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

$c = \pm \sqrt{- \frac{2}{5}}$

Step 3

Simplify $\pm \sqrt{- \frac{2}{5}}$ .

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Step 3.1

Rewrite $- 1$ as $i^{2}$ .

$c = \pm \sqrt{i^{2} \frac{2}{5}}$

Step 3.2

Pull terms out from under the radical.

$c = \pm i \sqrt{\frac{2}{5}}$

Step 3.3

Rewrite $\sqrt{\frac{2}{5}}$ as $\frac{\sqrt{2}}{\sqrt{5}}$ .

$c = \pm i \frac{\sqrt{2}}{\sqrt{5}}$

Step 3.4

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$c = \pm i (\frac{\sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$c = \pm i \frac{\sqrt{2} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 3.5.2

Raise $\sqrt{5}$ to the power of $1$ .

$c = \pm i \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 3.5.3

Raise $\sqrt{5}$ to the power of $1$ .

$c = \pm i \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 3.5.4

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

$c = \pm i \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 3.5.5

Add $1$ and $1$ .

$c = \pm i \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 3.5.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$c = \pm i \frac{\sqrt{2} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 3.5.6.2

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

$c = \pm i \frac{\sqrt{2} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 3.5.6.3

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

$c = \pm i \frac{\sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 3.5.6.4

Cancel the common factor of $2$ .

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Step 3.5.6.4.1

Cancel the common factor.

$c = \pm i \frac{\sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 3.5.6.4.2

Rewrite the expression.

$c = \pm i \frac{\sqrt{2} \sqrt{5}}{5^{1}}$

Step 3.5.6.5

Evaluate the exponent.

$c = \pm i \frac{\sqrt{2} \sqrt{5}}{5}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Combine using the product rule for radicals.

$c = \pm i \frac{\sqrt{2 \cdot 5}}{5}$

Step 3.6.2

Multiply $2$ by $5$ .

$c = \pm i \frac{\sqrt{10}}{5}$

Step 3.7

Combine $i$ and $\frac{\sqrt{10}}{5}$ .

$c = \pm \frac{i \sqrt{10}}{5}$

Step 4

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

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Step 4.1

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

$c = \frac{i \sqrt{10}}{5}$

Step 4.2

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

$c = - \frac{i \sqrt{10}}{5}$

Step 4.3

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

$c = \frac{i \sqrt{10}}{5}, - \frac{i \sqrt{10}}{5}$