Basic Math Examples

$\frac{2}{5} q \cdot (- \frac{1}{5}) \cdot q = 9$

Step 1

Simplify.

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Step 1.1

Rewrite $- 1 (\frac{1}{5})$ as $- (\frac{1}{5})$ .

$\frac{2}{5} q \cdot (- (\frac{1}{5})) \cdot q = 9$

Step 1.2

Combine $\frac{2}{5}$ and $q$ .

$\frac{2 q}{5} \cdot (- \frac{1}{5}) \cdot q = 9$

Step 1.3

Multiply $\frac{2 q}{5}$ by $\frac{1}{5}$ .

$- \frac{2 q}{5 \cdot 5} \cdot q = 9$

Step 1.4

Multiply $5$ by $5$ .

$- \frac{2 q}{25} \cdot q = 9$

Step 1.5

Combine $q$ and $\frac{2 q}{25}$ .

$- \frac{q (2 q)}{25} = 9$

Step 1.6

Raise $q$ to the power of $1$ .

$- \frac{2 (q^{1} q)}{25} = 9$

Step 1.7

Raise $q$ to the power of $1$ .

$- \frac{2 (q^{1} q^{1})}{25} = 9$

Step 1.8

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

$- \frac{2 q^{1 + 1}}{25} = 9$

Step 1.9

Add $1$ and $1$ .

$- \frac{2 q^{2}}{25} = 9$

Step 2

Divide each term in $- \frac{2 q^{2}}{25} = 9$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- \frac{2 q^{2}}{25} = 9$ by $- 1$ .

$\frac{- \frac{2 q^{2}}{25}}{- 1} = \frac{9}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{2 q^{2}}{25}}{1} = \frac{9}{- 1}$

Step 2.2.2

Divide $\frac{2 q^{2}}{25}$ by $1$ .

$\frac{2 q^{2}}{25} = \frac{9}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $9$ by $- 1$ .

$\frac{2 q^{2}}{25} = - 9$

Step 3

Multiply both sides of the equation by $\frac{25}{2}$ .

$\frac{25}{2} \cdot \frac{2 q^{2}}{25} = \frac{25}{2} \cdot - 9$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{25}{2} \cdot \frac{2 q^{2}}{25}$ .

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Step 4.1.1.1

Combine.

$\frac{25 (2 q^{2})}{2 \cdot 25} = \frac{25}{2} \cdot - 9$

Step 4.1.1.2

Cancel the common factor of $25$ .

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Step 4.1.1.2.1

Cancel the common factor.

$\frac{25 (2 q^{2})}{2 \cdot 25} = \frac{25}{2} \cdot - 9$

Step 4.1.1.2.2

Rewrite the expression.

$\frac{2 q^{2}}{2} = \frac{25}{2} \cdot - 9$

Step 4.1.1.3

Cancel the common factor of $2$ .

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Step 4.1.1.3.1

Cancel the common factor.

$\frac{2 q^{2}}{2} = \frac{25}{2} \cdot - 9$

Step 4.1.1.3.2

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

$q^{2} = \frac{25}{2} \cdot - 9$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{25}{2} \cdot - 9$ .

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Step 4.2.1.1

Multiply $\frac{25}{2} \cdot - 9$ .

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Step 4.2.1.1.1

Combine $\frac{25}{2}$ and $- 9$ .

$q^{2} = \frac{25 \cdot - 9}{2}$

Step 4.2.1.1.2

Multiply $25$ by $- 9$ .

$q^{2} = \frac{- 225}{2}$

Step 4.2.1.2

Move the negative in front of the fraction.

$q^{2} = - \frac{225}{2}$

Step 5

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

$q = \pm \sqrt{- \frac{225}{2}}$

Step 6

Simplify $\pm \sqrt{- \frac{225}{2}}$ .

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Step 6.1

Rewrite $- \frac{225}{2}$ as $i^{2} \frac{15^{2}}{2}$ .

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Step 6.1.1

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

$q = \pm \sqrt{i^{2} \frac{225}{2}}$

Step 6.1.2

Rewrite $225$ as $15^{2}$ .

$q = \pm \sqrt{i^{2} \frac{15^{2}}{2}}$

Step 6.2

Pull terms out from under the radical.

$q = \pm i \sqrt{\frac{15^{2}}{2}}$

Step 6.3

Raise $15$ to the power of $2$ .

$q = \pm i \sqrt{\frac{225}{2}}$

Step 6.4

Rewrite $\sqrt{\frac{225}{2}}$ as $\frac{\sqrt{225}}{\sqrt{2}}$ .

$q = \pm i \frac{\sqrt{225}}{\sqrt{2}}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Rewrite $225$ as $15^{2}$ .

$q = \pm i \frac{\sqrt{15^{2}}}{\sqrt{2}}$

Step 6.5.2

Pull terms out from under the radical, assuming positive real numbers.

$q = \pm i \frac{15}{\sqrt{2}}$

Step 6.6

Multiply $\frac{15}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$q = \pm i (\frac{15}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 6.7

Combine and simplify the denominator.

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Step 6.7.1

Multiply $\frac{15}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$q = \pm i \frac{15 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 6.7.2

Raise $\sqrt{2}$ to the power of $1$ .

$q = \pm i \frac{15 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 6.7.3

Raise $\sqrt{2}$ to the power of $1$ .

$q = \pm i \frac{15 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 6.7.4

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

$q = \pm i \frac{15 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 6.7.5

Add $1$ and $1$ .

$q = \pm i \frac{15 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 6.7.6

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

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Step 6.7.6.1

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

$q = \pm i \frac{15 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 6.7.6.2

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

$q = \pm i \frac{15 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 6.7.6.3

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

$q = \pm i \frac{15 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 6.7.6.4

Cancel the common factor of $2$ .

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Step 6.7.6.4.1

Cancel the common factor.

$q = \pm i \frac{15 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 6.7.6.4.2

Rewrite the expression.

$q = \pm i \frac{15 \sqrt{2}}{2^{1}}$

Step 6.7.6.5

Evaluate the exponent.

$q = \pm i \frac{15 \sqrt{2}}{2}$

Step 6.8

Combine $i$ and $\frac{15 \sqrt{2}}{2}$ .

$q = \pm \frac{i (15 \sqrt{2})}{2}$

Step 6.9

Move $15$ to the left of $i$ .

$q = \pm \frac{15 i \sqrt{2}}{2}$

Step 7

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

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Step 7.1

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

$q = \frac{15 i \sqrt{2}}{2}$

Step 7.2

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

$q = - \frac{15 i \sqrt{2}}{2}$

Step 7.3

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

$q = \frac{15 i \sqrt{2}}{2}, - \frac{15 i \sqrt{2}}{2}$