Basic Math Examples

$100000 = \frac{1}{2} \cdot ((2850) v^{2})$

Step 1

Rewrite the equation as $\frac{1}{2} \cdot (2850 v^{2}) = 100000$ .

$\frac{1}{2} \cdot (2850 v^{2}) = 100000$

Step 2

Multiply both sides of the equation by $\frac{1}{\frac{1}{2} \cdot 2850}$ .

$\frac{1}{\frac{1}{2} \cdot 2850} (\frac{1}{2} \cdot (2850 v^{2})) = \frac{1}{\frac{1}{2} \cdot 2850} \cdot 100000$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{1}{\frac{1}{2} \cdot 2850} (\frac{1}{2} \cdot (2850 v^{2}))$ .

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Step 3.1.1.1

Combine $\frac{1}{2}$ and $2850$ .

$\frac{1}{\frac{2850}{2}} (\frac{1}{2} \cdot (2850 v^{2})) = \frac{1}{\frac{1}{2} \cdot 2850} \cdot 100000$

Step 3.1.1.2

Divide $2850$ by $2$ .

$\frac{1}{1425} (\frac{1}{2} \cdot (2850 v^{2})) = \frac{1}{\frac{1}{2} \cdot 2850} \cdot 100000$

Step 3.1.1.3

Cancel the common factor of $1425$ .

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Step 3.1.1.3.1

Factor $1425$ out of $\frac{1}{2} \cdot (2850 v^{2})$ .

$\frac{1}{1425} (1425 (\frac{1}{2} \cdot (2 v^{2}))) = \frac{1}{\frac{1}{2} \cdot 2850} \cdot 100000$

Step 3.1.1.3.2

Cancel the common factor.

$\frac{1}{1425} (1425 (\frac{1}{2} \cdot (2 v^{2}))) = \frac{1}{\frac{1}{2} \cdot 2850} \cdot 100000$

Step 3.1.1.3.3

Rewrite the expression.

$\frac{1}{2} \cdot (2 v^{2}) = \frac{1}{\frac{1}{2} \cdot 2850} \cdot 100000$

Step 3.1.1.4

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

$\frac{2}{2} \cdot v^{2} = \frac{1}{\frac{1}{2} \cdot 2850} \cdot 100000$

Step 3.1.1.5

Combine $\frac{2}{2}$ and $v^{2}$ .

$\frac{2 v^{2}}{2} = \frac{1}{\frac{1}{2} \cdot 2850} \cdot 100000$

Step 3.1.1.6

Cancel the common factor of $2$ .

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Step 3.1.1.6.1

Cancel the common factor.

$\frac{2 v^{2}}{2} = \frac{1}{\frac{1}{2} \cdot 2850} \cdot 100000$

Step 3.1.1.6.2

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

$v^{2} = \frac{1}{\frac{1}{2} \cdot 2850} \cdot 100000$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{1}{\frac{1}{2} \cdot 2850} \cdot 100000$ .

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Step 3.2.1.1

Combine $\frac{1}{2}$ and $2850$ .

$v^{2} = \frac{1}{\frac{2850}{2}} \cdot 100000$

Step 3.2.1.2

Divide $2850$ by $2$ .

$v^{2} = \frac{1}{1425} \cdot 100000$

Step 3.2.1.3

Cancel the common factor of $25$ .

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Step 3.2.1.3.1

Factor $25$ out of $1425$ .

$v^{2} = \frac{1}{25 (57)} \cdot 100000$

Step 3.2.1.3.2

Factor $25$ out of $100000$ .

$v^{2} = \frac{1}{25 \cdot 57} \cdot (25 \cdot 4000)$

Step 3.2.1.3.3

Cancel the common factor.

$v^{2} = \frac{1}{25 \cdot 57} \cdot (25 \cdot 4000)$

Step 3.2.1.3.4

Rewrite the expression.

$v^{2} = \frac{1}{57} \cdot 4000$

Step 3.2.1.4

Combine $\frac{1}{57}$ and $4000$ .

$v^{2} = \frac{4000}{57}$

Step 4

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

$v = \pm \sqrt{\frac{4000}{57}}$

Step 5

Simplify $\pm \sqrt{\frac{4000}{57}}$ .

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Step 5.1

Rewrite $\sqrt{\frac{4000}{57}}$ as $\frac{\sqrt{4000}}{\sqrt{57}}$ .

$v = \pm \frac{\sqrt{4000}}{\sqrt{57}}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Rewrite $4000$ as $20^{2} \cdot 10$ .

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Step 5.2.1.1

Factor $400$ out of $4000$ .

$v = \pm \frac{\sqrt{400 (10)}}{\sqrt{57}}$

Step 5.2.1.2

Rewrite $400$ as $20^{2}$ .

$v = \pm \frac{\sqrt{20^{2} \cdot 10}}{\sqrt{57}}$

Step 5.2.2

Pull terms out from under the radical.

$v = \pm \frac{20 \sqrt{10}}{\sqrt{57}}$

Step 5.3

Multiply $\frac{20 \sqrt{10}}{\sqrt{57}}$ by $\frac{\sqrt{57}}{\sqrt{57}}$ .

$v = \pm \frac{20 \sqrt{10}}{\sqrt{57}} \cdot \frac{\sqrt{57}}{\sqrt{57}}$

Step 5.4

Combine and simplify the denominator.

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Step 5.4.1

Multiply $\frac{20 \sqrt{10}}{\sqrt{57}}$ by $\frac{\sqrt{57}}{\sqrt{57}}$ .

$v = \pm \frac{20 \sqrt{10} \sqrt{57}}{\sqrt{57} \sqrt{57}}$

Step 5.4.2

Raise $\sqrt{57}$ to the power of $1$ .

$v = \pm \frac{20 \sqrt{10} \sqrt{57}}{{\sqrt{57}}^{1} \sqrt{57}}$

Step 5.4.3

Raise $\sqrt{57}$ to the power of $1$ .

$v = \pm \frac{20 \sqrt{10} \sqrt{57}}{{\sqrt{57}}^{1} {\sqrt{57}}^{1}}$

Step 5.4.4

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

$v = \pm \frac{20 \sqrt{10} \sqrt{57}}{{\sqrt{57}}^{1 + 1}}$

Step 5.4.5

Add $1$ and $1$ .

$v = \pm \frac{20 \sqrt{10} \sqrt{57}}{{\sqrt{57}}^{2}}$

Step 5.4.6

Rewrite ${\sqrt{57}}^{2}$ as $57$ .

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Step 5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$v = \pm \frac{20 \sqrt{10} \sqrt{57}}{{(57^{\frac{1}{2}})}^{2}}$

Step 5.4.6.2

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

$v = \pm \frac{20 \sqrt{10} \sqrt{57}}{57^{\frac{1}{2} \cdot 2}}$

Step 5.4.6.3

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

$v = \pm \frac{20 \sqrt{10} \sqrt{57}}{57^{\frac{2}{2}}}$

Step 5.4.6.4

Cancel the common factor of $2$ .

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Step 5.4.6.4.1

Cancel the common factor.

$v = \pm \frac{20 \sqrt{10} \sqrt{57}}{57^{\frac{2}{2}}}$

Step 5.4.6.4.2

Rewrite the expression.

$v = \pm \frac{20 \sqrt{10} \sqrt{57}}{57^{1}}$

Step 5.4.6.5

Evaluate the exponent.

$v = \pm \frac{20 \sqrt{10} \sqrt{57}}{57}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Combine using the product rule for radicals.

$v = \pm \frac{20 \sqrt{57 \cdot 10}}{57}$

Step 5.5.2

Multiply $57$ by $10$ .

$v = \pm \frac{20 \sqrt{570}}{57}$

Step 6

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

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Step 6.1

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

$v = \frac{20 \sqrt{570}}{57}$

Step 6.2

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

$v = - \frac{20 \sqrt{570}}{57}$

Step 6.3

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

$v = \frac{20 \sqrt{570}}{57}, - \frac{20 \sqrt{570}}{57}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$v = \frac{20 \sqrt{570}}{57}, - \frac{20 \sqrt{570}}{57}$

Decimal Form:

$v = 8.37707816 \dots, - 8.37707816 \dots$