Finite Math Examples

$σ^{- 3} + 9.96 σ^{- 2} + 384.47 σ - 2095.37 = 0$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{σ^{3}} + 9.96 σ^{- 2} + 384.47 σ - 2095.37 = 0$

Step 2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{σ^{3}} + 9.96 (\frac{1}{σ^{2}}) + 384.47 σ - 2095.37 = 0$

Step 3

Combine $9.96$ and $\frac{1}{σ^{2}}$ .

$\frac{1}{σ^{3}} + \frac{9.96}{σ^{2}} + 384.47 σ - 2095.37 = 0$

Step 4

Reorder terms.

$384.47 σ + \frac{9.96}{σ^{2}} + \frac{1}{σ^{3}} - 2095.37 = 0$

Step 5

Factor $0.01$ out of $384.47 σ + \frac{9.96}{σ^{2}} + \frac{1}{σ^{3}} - 2095.37$ .

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Step 5.1

Factor $0.01$ out of $384.47 σ$ .

$0.01 (38447 σ) + \frac{9.96}{σ^{2}} + \frac{1}{σ^{3}} - 2095.37 = 0$

Step 5.2

Factor $0.01$ out of $\frac{9.96}{σ^{2}}$ .

$0.01 (38447 σ) + 0.01 (\frac{996}{σ^{2}}) + \frac{1}{σ^{3}} - 2095.37 = 0$

Step 5.3

Factor $0.01$ out of $\frac{1}{σ^{3}}$ .

$0.01 (38447 σ) + 0.01 (\frac{996}{σ^{2}}) + 0.01 (\frac{100}{σ^{3}}) - 2095.37 = 0$

Step 5.4

Factor $0.01$ out of $- 2095.37$ .

$0.01 (38447 σ) + 0.01 (\frac{996}{σ^{2}}) + 0.01 (\frac{100}{σ^{3}}) + 0.01 (- 209537) = 0$

Step 5.5

Factor $0.01$ out of $0.01 (38447 σ) + 0.01 \frac{996}{σ^{2}}$ .

$0.01 (38447 σ + \frac{996}{σ^{2}}) + 0.01 (\frac{100}{σ^{3}}) + 0.01 (- 209537) = 0$

Step 5.6

Factor $0.01$ out of $0.01 (38447 σ + \frac{996}{σ^{2}}) + 0.01 \frac{100}{σ^{3}}$ .

$0.01 (38447 σ + \frac{996}{σ^{2}} + \frac{100}{σ^{3}}) + 0.01 (- 209537) = 0$

Step 5.7

Factor $0.01$ out of $0.01 (38447 σ + \frac{996}{σ^{2}} + \frac{100}{σ^{3}}) + 0.01 (- 209537)$ .

$0.01 (38447 σ + \frac{996}{σ^{2}} + \frac{100}{σ^{3}} - 209537) = 0$

Step 6

To write $38447 σ$ as a fraction with a common denominator, multiply by $\frac{σ^{2}}{σ^{2}}$ .

$0.01 (\frac{38447 σ \cdot σ^{2}}{σ^{2}} + \frac{996}{σ^{2}} + \frac{100}{σ^{3}} - 209537) = 0$

Step 7

Combine the numerators over the common denominator.

$0.01 (\frac{38447 σ \cdot σ^{2} + 996}{σ^{2}} + \frac{100}{σ^{3}} - 209537) = 0$

Step 8

Multiply $σ$ by $σ^{2}$ by adding the exponents.

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Step 8.1

Move $σ^{2}$ .

$0.01 (\frac{38447 (σ^{2} σ) + 996}{σ^{2}} + \frac{100}{σ^{3}} - 209537) = 0$

Step 8.2

Multiply $σ^{2}$ by $σ$ .

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Step 8.2.1

Raise $σ$ to the power of $1$ .

$0.01 (\frac{38447 (σ^{2} σ) + 996}{σ^{2}} + \frac{100}{σ^{3}} - 209537) = 0$

Step 8.2.2

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

$0.01 (\frac{38447 σ^{2 + 1} + 996}{σ^{2}} + \frac{100}{σ^{3}} - 209537) = 0$

Step 8.3

Add $2$ and $1$ .

$0.01 (\frac{38447 σ^{3} + 996}{σ^{2}} + \frac{100}{σ^{3}} - 209537) = 0$

Step 9

To write $\frac{38447 σ^{3} + 996}{σ^{2}}$ as a fraction with a common denominator, multiply by $\frac{σ}{σ}$ .

$0.01 (\frac{38447 σ^{3} + 996}{σ^{2}} \cdot \frac{σ}{σ} + \frac{100}{σ^{3}} - 209537) = 0$

Step 10

Write each expression with a common denominator of $σ^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.1

Multiply $\frac{38447 σ^{3} + 996}{σ^{2}}$ by $\frac{σ}{σ}$ .

$0.01 (\frac{(38447 σ^{3} + 996) σ}{σ^{2} σ} + \frac{100}{σ^{3}} - 209537) = 0$

Step 10.2

Multiply $σ^{2}$ by $σ$ by adding the exponents.

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Step 10.2.1

Multiply $σ^{2}$ by $σ$ .

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Step 10.2.1.1

Raise $σ$ to the power of $1$ .

$0.01 (\frac{(38447 σ^{3} + 996) σ}{σ^{2} σ} + \frac{100}{σ^{3}} - 209537) = 0$

Step 10.2.1.2

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

$0.01 (\frac{(38447 σ^{3} + 996) σ}{σ^{2 + 1}} + \frac{100}{σ^{3}} - 209537) = 0$

Step 10.2.2

Add $2$ and $1$ .

$0.01 (\frac{(38447 σ^{3} + 996) σ}{σ^{3}} + \frac{100}{σ^{3}} - 209537) = 0$

Step 11

Combine the numerators over the common denominator.

$0.01 (\frac{(38447 σ^{3} + 996) σ + 100}{σ^{3}} - 209537) = 0$

Step 12

Simplify the numerator.

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Step 12.1

Apply the distributive property.

$0.01 (\frac{38447 σ^{3} σ + 996 σ + 100}{σ^{3}} - 209537) = 0$

Step 12.2

Multiply $σ^{3}$ by $σ$ by adding the exponents.

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Step 12.2.1

Move $σ$ .

$0.01 (\frac{38447 (σ \cdot σ^{3}) + 996 σ + 100}{σ^{3}} - 209537) = 0$

Step 12.2.2

Multiply $σ$ by $σ^{3}$ .

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Step 12.2.2.1

Raise $σ$ to the power of $1$ .

$0.01 (\frac{38447 (σ \cdot σ^{3}) + 996 σ + 100}{σ^{3}} - 209537) = 0$

Step 12.2.2.2

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

$0.01 (\frac{38447 σ^{1 + 3} + 996 σ + 100}{σ^{3}} - 209537) = 0$

Step 12.2.3

Add $1$ and $3$ .

$0.01 (\frac{38447 σ^{4} + 996 σ + 100}{σ^{3}} - 209537) = 0$

Step 13

To write $- 209537$ as a fraction with a common denominator, multiply by $\frac{σ^{3}}{σ^{3}}$ .

$0.01 (\frac{38447 σ^{4} + 996 σ + 100}{σ^{3}} - 209537 \cdot \frac{σ^{3}}{σ^{3}}) = 0$

Step 14

Combine $- 209537$ and $\frac{σ^{3}}{σ^{3}}$ .

$0.01 (\frac{38447 σ^{4} + 996 σ + 100}{σ^{3}} + \frac{- 209537 σ^{3}}{σ^{3}}) = 0$

Step 15

Combine the numerators over the common denominator.

$0.01 (\frac{38447 σ^{4} + 996 σ + 100 - 209537 σ^{3}}{σ^{3}}) = 0$

Step 16

Reorder terms.

$0.01 (\frac{38447 σ^{4} - 209537 σ^{3} + 996 σ + 100}{σ^{3}}) = 0$

Step 17

Combine $0.01$ and $\frac{38447 σ^{4} - 209537 σ^{3} + 996 σ + 100}{σ^{3}}$ .

$\frac{0.01 (38447 σ^{4} - 209537 σ^{3} + 996 σ + 100)}{σ^{3}} = 0$

Step 18

Graph each side of the equation. The solution is the x-value of the point of intersection.

$σ \approx 0.09879687, 5.44913358$

Step 19