Finite Math Examples

$1296 - \frac{27}{25000} l^{2} + \frac{171}{100000} - 0.14 = 0$

Step 1

Simplify $1296 - \frac{27}{25000} l^{2} + \frac{171}{100000} - 0.14$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $l^{2}$ and $\frac{27}{25000}$ .

$1296 - \frac{l^{2} \cdot 27}{25000} + \frac{171}{100000} - 0.14 = 0$

Step 1.1.2

Move $27$ to the left of $l^{2}$ .

$1296 - \frac{27 l^{2}}{25000} + \frac{171}{100000} - 0.14 = 0$

Step 1.2

To write $1296$ as a fraction with a common denominator, multiply by $\frac{100000}{100000}$ .

$- \frac{27 l^{2}}{25000} + 1296 \cdot \frac{100000}{100000} + \frac{171}{100000} - 0.14 = 0$

Step 1.3

Combine $1296$ and $\frac{100000}{100000}$ .

$- \frac{27 l^{2}}{25000} + \frac{1296 \cdot 100000}{100000} + \frac{171}{100000} - 0.14 = 0$

Step 1.4

Combine the numerators over the common denominator.

$- \frac{27 l^{2}}{25000} + \frac{1296 \cdot 100000 + 171}{100000} - 0.14 = 0$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $1296$ by $100000$ .

$- \frac{27 l^{2}}{25000} + \frac{129600000 + 171}{100000} - 0.14 = 0$

Step 1.5.2

Add $129600000$ and $171$ .

$- \frac{27 l^{2}}{25000} + \frac{129600171}{100000} - 0.14 = 0$

Step 1.6

To write $- 0.14$ as a fraction with a common denominator, multiply by $\frac{100000}{100000}$ .

$- \frac{27 l^{2}}{25000} + \frac{129600171}{100000} - 0.14 \cdot \frac{100000}{100000} = 0$

Step 1.7

Combine $- 0.14$ and $\frac{100000}{100000}$ .

$- \frac{27 l^{2}}{25000} + \frac{129600171}{100000} + \frac{- 0.14 \cdot 100000}{100000} = 0$

Step 1.8

Combine the numerators over the common denominator.

$- \frac{27 l^{2}}{25000} + \frac{129600171 - 0.14 \cdot 100000}{100000} = 0$

Step 1.9

Simplify the numerator.

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Step 1.9.1

Multiply $- 0.14$ by $100000$ .

$- \frac{27 l^{2}}{25000} + \frac{129600171 - 14000}{100000} = 0$

Step 1.9.2

Subtract $14000$ from $129600171$ .

$- \frac{27 l^{2}}{25000} + \frac{129586171}{100000} = 0$

Step 1.10

Cancel the common factor of $129586171$ and $100000$ .

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Step 1.10.1

Rewrite $129586171$ as $1 (129586171)$ .

$- \frac{27 l^{2}}{25000} + \frac{1 (129586171)}{100000} = 0$

Step 1.10.2

Cancel the common factors.

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Step 1.10.2.1

Rewrite $100000$ as $1 (100000)$ .

$- \frac{27 l^{2}}{25000} + \frac{1 \cdot 129586171}{1 \cdot 100000} = 0$

Step 1.10.2.2

Cancel the common factor.

$- \frac{27 l^{2}}{25000} + \frac{1 \cdot 129586171}{1 \cdot 100000} = 0$

Step 1.10.2.3

Rewrite the expression.

$- \frac{27 l^{2}}{25000} + \frac{129586171}{100000} = 0$

Step 2

Subtract $\frac{129586171}{100000}$ from both sides of the equation.

$- \frac{27 l^{2}}{25000} = - \frac{129586171}{100000}$

Step 3

Multiply both sides of the equation by $- \frac{25000}{27}$ .

$- \frac{25000}{27} (- \frac{27 l^{2}}{25000}) = - \frac{25000}{27} (- \frac{129586171}{100000})$

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 $- \frac{25000}{27} (- \frac{27 l^{2}}{25000})$ .

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Step 4.1.1.1

Cancel the common factor of $25000$ .

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Step 4.1.1.1.1

Move the leading negative in $- \frac{25000}{27}$ into the numerator.

$\frac{- 25000}{27} (- \frac{27 l^{2}}{25000}) = - \frac{25000}{27} (- \frac{129586171}{100000})$

Step 4.1.1.1.2

Move the leading negative in $- \frac{27 l^{2}}{25000}$ into the numerator.

$\frac{- 25000}{27} \cdot \frac{- 27 l^{2}}{25000} = - \frac{25000}{27} (- \frac{129586171}{100000})$

Step 4.1.1.1.3

Factor $25000$ out of $- 25000$ .

$\frac{25000 (- 1)}{27} \cdot \frac{- 27 l^{2}}{25000} = - \frac{25000}{27} (- \frac{129586171}{100000})$

Step 4.1.1.1.4

Cancel the common factor.

$\frac{25000 \cdot - 1}{27} \cdot \frac{- 27 l^{2}}{25000} = - \frac{25000}{27} (- \frac{129586171}{100000})$

Step 4.1.1.1.5

Rewrite the expression.

$\frac{- 1}{27} (- 27 l^{2}) = - \frac{25000}{27} (- \frac{129586171}{100000})$

Step 4.1.1.2

Cancel the common factor of $27$ .

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Step 4.1.1.2.1

Factor $27$ out of $- 27 l^{2}$ .

$\frac{- 1}{27} (27 (- l^{2})) = - \frac{25000}{27} (- \frac{129586171}{100000})$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{- 1}{27} (27 (- l^{2})) = - \frac{25000}{27} (- \frac{129586171}{100000})$

Step 4.1.1.2.3

Rewrite the expression.

$- - l^{2} = - \frac{25000}{27} (- \frac{129586171}{100000})$

Step 4.1.1.3

Multiply.

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Step 4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 l^{2} = - \frac{25000}{27} (- \frac{129586171}{100000})$

Step 4.1.1.3.2

Multiply $l^{2}$ by $1$ .

$l^{2} = - \frac{25000}{27} (- \frac{129586171}{100000})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{25000}{27} (- \frac{129586171}{100000})$ .

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Step 4.2.1.1

Cancel the common factor of $25000$ .

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Step 4.2.1.1.1

Move the leading negative in $- \frac{25000}{27}$ into the numerator.

$l^{2} = \frac{- 25000}{27} (- \frac{129586171}{100000})$

Step 4.2.1.1.2

Move the leading negative in $- \frac{129586171}{100000}$ into the numerator.

$l^{2} = \frac{- 25000}{27} \cdot \frac{- 129586171}{100000}$

Step 4.2.1.1.3

Factor $25000$ out of $- 25000$ .

$l^{2} = \frac{25000 (- 1)}{27} \cdot \frac{- 129586171}{100000}$

Step 4.2.1.1.4

Factor $25000$ out of $100000$ .

$l^{2} = \frac{25000 \cdot - 1}{27} \cdot \frac{- 129586171}{25000 \cdot 4}$

Step 4.2.1.1.5

Cancel the common factor.

$l^{2} = \frac{25000 \cdot - 1}{27} \cdot \frac{- 129586171}{25000 \cdot 4}$

Step 4.2.1.1.6

Rewrite the expression.

$l^{2} = \frac{- 1}{27} \cdot \frac{- 129586171}{4}$

Step 4.2.1.2

Multiply $\frac{- 1}{27}$ by $\frac{- 129586171}{4}$ .

$l^{2} = \frac{- - 129586171}{27 \cdot 4}$

Step 4.2.1.3

Multiply.

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Step 4.2.1.3.1

Multiply $- 1$ by $- 129586171$ .

$l^{2} = \frac{129586171}{27 \cdot 4}$

Step 4.2.1.3.2

Multiply $27$ by $4$ .

$l^{2} = \frac{129586171}{108}$

Step 5

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

$l = \pm \sqrt{\frac{129586171}{108}}$

Step 6

Simplify $\pm \sqrt{\frac{129586171}{108}}$ .

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Step 6.1

Rewrite $\sqrt{\frac{129586171}{108}}$ as $\frac{\sqrt{129586171}}{\sqrt{108}}$ .

$l = \pm \frac{\sqrt{129586171}}{\sqrt{108}}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Rewrite $108$ as $6^{2} \cdot 3$ .

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Step 6.2.1.1

Factor $36$ out of $108$ .

$l = \pm \frac{\sqrt{129586171}}{\sqrt{36 (3)}}$

Step 6.2.1.2

Rewrite $36$ as $6^{2}$ .

$l = \pm \frac{\sqrt{129586171}}{\sqrt{6^{2} \cdot 3}}$

Step 6.2.2

Pull terms out from under the radical.

$l = \pm \frac{\sqrt{129586171}}{6 \sqrt{3}}$

Step 6.3

Multiply $\frac{\sqrt{129586171}}{6 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$l = \pm \frac{\sqrt{129586171}}{6 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 6.4

Combine and simplify the denominator.

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Step 6.4.1

Multiply $\frac{\sqrt{129586171}}{6 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$l = \pm \frac{\sqrt{129586171} \sqrt{3}}{6 \sqrt{3} \sqrt{3}}$

Step 6.4.2

Move $\sqrt{3}$ .

$l = \pm \frac{\sqrt{129586171} \sqrt{3}}{6 (\sqrt{3} \sqrt{3})}$

Step 6.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$l = \pm \frac{\sqrt{129586171} \sqrt{3}}{6 ({\sqrt{3}}^{1} \sqrt{3})}$

Step 6.4.4

Raise $\sqrt{3}$ to the power of $1$ .

$l = \pm \frac{\sqrt{129586171} \sqrt{3}}{6 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

Step 6.4.5

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

$l = \pm \frac{\sqrt{129586171} \sqrt{3}}{6 {\sqrt{3}}^{1 + 1}}$

Step 6.4.6

Add $1$ and $1$ .

$l = \pm \frac{\sqrt{129586171} \sqrt{3}}{6 {\sqrt{3}}^{2}}$

Step 6.4.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$l = \pm \frac{\sqrt{129586171} \sqrt{3}}{6 {(3^{\frac{1}{2}})}^{2}}$

Step 6.4.7.2

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

$l = \pm \frac{\sqrt{129586171} \sqrt{3}}{6 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 6.4.7.3

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

$l = \pm \frac{\sqrt{129586171} \sqrt{3}}{6 \cdot 3^{\frac{2}{2}}}$

Step 6.4.7.4

Cancel the common factor of $2$ .

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Step 6.4.7.4.1

Cancel the common factor.

$l = \pm \frac{\sqrt{129586171} \sqrt{3}}{6 \cdot 3^{\frac{2}{2}}}$

Step 6.4.7.4.2

Rewrite the expression.

$l = \pm \frac{\sqrt{129586171} \sqrt{3}}{6 \cdot 3^{1}}$

Step 6.4.7.5

Evaluate the exponent.

$l = \pm \frac{\sqrt{129586171} \sqrt{3}}{6 \cdot 3}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Combine using the product rule for radicals.

$l = \pm \frac{\sqrt{129586171 \cdot 3}}{6 \cdot 3}$

Step 6.5.2

Multiply $129586171$ by $3$ .

$l = \pm \frac{\sqrt{388758513}}{6 \cdot 3}$

Step 6.6

Multiply $6$ by $3$ .

$l = \pm \frac{\sqrt{388758513}}{18}$

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.

$l = \frac{\sqrt{388758513}}{18}$

Step 7.2

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

$l = - \frac{\sqrt{388758513}}{18}$

Step 7.3

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

$l = \frac{\sqrt{388758513}}{18}, - \frac{\sqrt{388758513}}{18}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$l = \frac{\sqrt{388758513}}{18}, - \frac{\sqrt{388758513}}{18}$

Decimal Form:

$l = 1095.38666858 \dots, - 1095.38666858 \dots$