Finite Math Examples

$\frac{180 - \frac{15 (15 + 15 + 1)}{2}}{\sqrt{\frac{15 (15) (15 + 15 + 1)}{12}}}$

Step 1

Cancel the common factor of $15$ and $12$ .

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Step 1.1

Factor $3$ out of $15 (15) (15 + 15 + 1)$ .

$\frac{180 - \frac{15 (15 + 15 + 1)}{2}}{\sqrt{\frac{3 (5 \cdot 15 (15 + 15 + 1))}{12}}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $3$ out of $12$ .

$\frac{180 - \frac{15 (15 + 15 + 1)}{2}}{\sqrt{\frac{3 (5 \cdot 15 (15 + 15 + 1))}{3 (4)}}}$

Step 1.2.2

Cancel the common factor.

$\frac{180 - \frac{15 (15 + 15 + 1)}{2}}{\sqrt{\frac{3 (5 \cdot 15 (15 + 15 + 1))}{3 \cdot 4}}}$

Step 1.2.3

Rewrite the expression.

$\frac{180 - \frac{15 (15 + 15 + 1)}{2}}{\sqrt{\frac{5 \cdot 15 (15 + 15 + 1)}{4}}}$

Step 2

Simplify the numerator.

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Step 2.1

Add $15$ and $15$ .

$\frac{180 - \frac{15 (30 + 1)}{2}}{\sqrt{\frac{5 \cdot 15 (15 + 15 + 1)}{4}}}$

Step 2.2

Add $30$ and $1$ .

$\frac{180 - \frac{15 \cdot 31}{2}}{\sqrt{\frac{5 \cdot 15 (15 + 15 + 1)}{4}}}$

Step 3

Multiply.

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Step 3.1

Multiply $15$ by $31$ .

$\frac{180 - \frac{465}{2}}{\sqrt{\frac{5 \cdot 15 (15 + 15 + 1)}{4}}}$

Step 3.2

Multiply $5$ by $15$ .

$\frac{180 - \frac{465}{2}}{\sqrt{\frac{75 (15 + 15 + 1)}{4}}}$

Step 4

Simplify the numerator.

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Step 4.1

Add $15$ and $15$ .

$\frac{180 - \frac{465}{2}}{\sqrt{\frac{75 (30 + 1)}{4}}}$

Step 4.2

Add $30$ and $1$ .

$\frac{180 - \frac{465}{2}}{\sqrt{\frac{75 \cdot 31}{4}}}$

Step 5

Multiply $75$ by $31$ .

$\frac{180 - \frac{465}{2}}{\sqrt{\frac{2325}{4}}}$

Step 6

Simplify the numerator.

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Step 6.1

To write $180$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{180 \cdot \frac{2}{2} - \frac{465}{2}}{\sqrt{\frac{2325}{4}}}$

Step 6.2

Combine $180$ and $\frac{2}{2}$ .

$\frac{\frac{180 \cdot 2}{2} - \frac{465}{2}}{\sqrt{\frac{2325}{4}}}$

Step 6.3

Combine the numerators over the common denominator.

$\frac{\frac{180 \cdot 2 - 465}{2}}{\sqrt{\frac{2325}{4}}}$

Step 6.4

Simplify the numerator.

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Step 6.4.1

Multiply $180$ by $2$ .

$\frac{\frac{360 - 465}{2}}{\sqrt{\frac{2325}{4}}}$

Step 6.4.2

Subtract $465$ from $360$ .

$\frac{\frac{- 105}{2}}{\sqrt{\frac{2325}{4}}}$

Step 6.5

Move the negative in front of the fraction.

$\frac{- \frac{105}{2}}{\sqrt{\frac{2325}{4}}}$

Step 7

Simplify the denominator.

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Step 7.1

Rewrite $\sqrt{\frac{2325}{4}}$ as $\frac{\sqrt{2325}}{\sqrt{4}}$ .

$\frac{- \frac{105}{2}}{\frac{\sqrt{2325}}{\sqrt{4}}}$

Step 7.2

Simplify the numerator.

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Step 7.2.1

Rewrite $2325$ as $5^{2} \cdot 93$ .

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Step 7.2.1.1

Factor $25$ out of $2325$ .

$\frac{- \frac{105}{2}}{\frac{\sqrt{25 (93)}}{\sqrt{4}}}$

Step 7.2.1.2

Rewrite $25$ as $5^{2}$ .

$\frac{- \frac{105}{2}}{\frac{\sqrt{5^{2} \cdot 93}}{\sqrt{4}}}$

Step 7.2.2

Pull terms out from under the radical.

$\frac{- \frac{105}{2}}{\frac{5 \sqrt{93}}{\sqrt{4}}}$

Step 7.3

Simplify the denominator.

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Step 7.3.1

Rewrite $4$ as $2^{2}$ .

$\frac{- \frac{105}{2}}{\frac{5 \sqrt{93}}{\sqrt{2^{2}}}}$

Step 7.3.2

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

$\frac{- \frac{105}{2}}{\frac{5 \sqrt{93}}{2}}$

Step 8

Multiply the numerator by the reciprocal of the denominator.

$- \frac{105}{2} \cdot \frac{2}{5 \sqrt{93}}$

Step 9

Cancel the common factor of $5$ .

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Step 9.1

Move the leading negative in $- \frac{105}{2}$ into the numerator.

$\frac{- 105}{2} \cdot \frac{2}{5 \sqrt{93}}$

Step 9.2

Factor $5$ out of $- 105$ .

$\frac{5 (- 21)}{2} \cdot \frac{2}{5 \sqrt{93}}$

Step 9.3

Factor $5$ out of $5 \sqrt{93}$ .

$\frac{5 (- 21)}{2} \cdot \frac{2}{5 (\sqrt{93})}$

Step 9.4

Cancel the common factor.

$\frac{5 \cdot - 21}{2} \cdot \frac{2}{5 \sqrt{93}}$

Step 9.5

Rewrite the expression.

$\frac{- 21}{2} \cdot \frac{2}{\sqrt{93}}$

Step 10

Cancel the common factor of $2$ .

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Step 10.1

Cancel the common factor.

$\frac{- 21}{2} \cdot \frac{2}{\sqrt{93}}$

Step 10.2

Rewrite the expression.

$- 21 \frac{1}{\sqrt{93}}$

Step 11

Combine $- 21$ and $\frac{1}{\sqrt{93}}$ .

$\frac{- 21}{\sqrt{93}}$

Step 12

Move the negative in front of the fraction.

$- \frac{21}{\sqrt{93}}$

Step 13

Multiply $\frac{21}{\sqrt{93}}$ by $\frac{\sqrt{93}}{\sqrt{93}}$ .

$- (\frac{21}{\sqrt{93}} \cdot \frac{\sqrt{93}}{\sqrt{93}})$

Step 14

Combine and simplify the denominator.

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Step 14.1

Multiply $\frac{21}{\sqrt{93}}$ by $\frac{\sqrt{93}}{\sqrt{93}}$ .

$- \frac{21 \sqrt{93}}{\sqrt{93} \sqrt{93}}$

Step 14.2

Raise $\sqrt{93}$ to the power of $1$ .

$- \frac{21 \sqrt{93}}{{\sqrt{93}}^{1} \sqrt{93}}$

Step 14.3

Raise $\sqrt{93}$ to the power of $1$ .

$- \frac{21 \sqrt{93}}{{\sqrt{93}}^{1} {\sqrt{93}}^{1}}$

Step 14.4

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

$- \frac{21 \sqrt{93}}{{\sqrt{93}}^{1 + 1}}$

Step 14.5

Add $1$ and $1$ .

$- \frac{21 \sqrt{93}}{{\sqrt{93}}^{2}}$

Step 14.6

Rewrite ${\sqrt{93}}^{2}$ as $93$ .

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Step 14.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{93}$ as $93^{\frac{1}{2}}$ .

$- \frac{21 \sqrt{93}}{{(93^{\frac{1}{2}})}^{2}}$

Step 14.6.2

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

$- \frac{21 \sqrt{93}}{93^{\frac{1}{2} \cdot 2}}$

Step 14.6.3

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

$- \frac{21 \sqrt{93}}{93^{\frac{2}{2}}}$

Step 14.6.4

Cancel the common factor of $2$ .

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Step 14.6.4.1

Cancel the common factor.

$- \frac{21 \sqrt{93}}{93^{\frac{2}{2}}}$

Step 14.6.4.2

Rewrite the expression.

$- \frac{21 \sqrt{93}}{93^{1}}$

Step 14.6.5

Evaluate the exponent.

$- \frac{21 \sqrt{93}}{93}$

Step 15

Cancel the common factor of $21$ and $93$ .

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Step 15.1

Factor $3$ out of $21 \sqrt{93}$ .

$- \frac{3 (7 \sqrt{93})}{93}$

Step 15.2

Cancel the common factors.

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Step 15.2.1

Factor $3$ out of $93$ .

$- \frac{3 (7 \sqrt{93})}{3 (31)}$

Step 15.2.2

Cancel the common factor.

$- \frac{3 (7 \sqrt{93})}{3 \cdot 31}$

Step 15.2.3

Rewrite the expression.

$- \frac{7 \sqrt{93}}{31}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$- \frac{7 \sqrt{93}}{31}$

Decimal Form:

$- 2.17759855 \dots$