Finite Math Examples

$\frac{\sqrt{3}}{16} \cdot (1 \frac{6}{4} + {(\frac{3}{4})}^{2} + {(\frac{3}{4})}^{7})$

Step 1

Convert $1 \frac{6}{4}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

$\frac{\sqrt{3}}{16} \cdot (1 + \frac{6}{4} + {(\frac{3}{4})}^{2} + {(\frac{3}{4})}^{7})$

Step 1.2

Add $1$ and $\frac{6}{4}$ .

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Step 1.2.1

Write $1$ as a fraction with a common denominator.

$\frac{\sqrt{3}}{16} \cdot (\frac{4}{4} + \frac{6}{4} + {(\frac{3}{4})}^{2} + {(\frac{3}{4})}^{7})$

Step 1.2.2

Combine the numerators over the common denominator.

$\frac{\sqrt{3}}{16} \cdot (\frac{4 + 6}{4} + {(\frac{3}{4})}^{2} + {(\frac{3}{4})}^{7})$

Step 1.2.3

Add $4$ and $6$ .

$\frac{\sqrt{3}}{16} \cdot (\frac{10}{4} + {(\frac{3}{4})}^{2} + {(\frac{3}{4})}^{7})$

Step 2

Find the common denominator.

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Step 2.1

Write ${(\frac{3}{4})}^{2}$ as a fraction with denominator $1$ .

$\frac{\sqrt{3}}{16} \cdot (\frac{10}{4} + \frac{{(\frac{3}{4})}^{2}}{1} + {(\frac{3}{4})}^{7})$

Step 2.2

Multiply $\frac{{(\frac{3}{4})}^{2}}{1}$ by $\frac{4}{4}$ .

$\frac{\sqrt{3}}{16} \cdot (\frac{10}{4} + \frac{{(\frac{3}{4})}^{2}}{1} \cdot \frac{4}{4} + {(\frac{3}{4})}^{7})$

Step 2.3

Multiply $\frac{{(\frac{3}{4})}^{2}}{1}$ by $\frac{4}{4}$ .

$\frac{\sqrt{3}}{16} \cdot (\frac{10}{4} + \frac{{(\frac{3}{4})}^{2} \cdot 4}{4} + {(\frac{3}{4})}^{7})$

Step 2.4

Write ${(\frac{3}{4})}^{7}$ as a fraction with denominator $1$ .

$\frac{\sqrt{3}}{16} \cdot (\frac{10}{4} + \frac{{(\frac{3}{4})}^{2} \cdot 4}{4} + \frac{{(\frac{3}{4})}^{7}}{1})$

Step 2.5

Multiply $\frac{{(\frac{3}{4})}^{7}}{1}$ by $\frac{4}{4}$ .

$\frac{\sqrt{3}}{16} \cdot (\frac{10}{4} + \frac{{(\frac{3}{4})}^{2} \cdot 4}{4} + \frac{{(\frac{3}{4})}^{7}}{1} \cdot \frac{4}{4})$

Step 2.6

Multiply $\frac{{(\frac{3}{4})}^{7}}{1}$ by $\frac{4}{4}$ .

$\frac{\sqrt{3}}{16} \cdot (\frac{10}{4} + \frac{{(\frac{3}{4})}^{2} \cdot 4}{4} + \frac{{(\frac{3}{4})}^{7} \cdot 4}{4})$

Step 3

Combine the numerators over the common denominator.

$\frac{\sqrt{3}}{16} \cdot \frac{10 + {(\frac{3}{4})}^{2} \cdot 4 + {(\frac{3}{4})}^{7} \cdot 4}{4}$

Step 4

Simplify each term.

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Step 4.1

Apply the product rule to $\frac{3}{4}$ .

$\frac{\sqrt{3}}{16} \cdot \frac{10 + \frac{3^{2}}{4^{2}} \cdot 4 + {(\frac{3}{4})}^{7} \cdot 4}{4}$

Step 4.2

Raise $3$ to the power of $2$ .

$\frac{\sqrt{3}}{16} \cdot \frac{10 + \frac{9}{4^{2}} \cdot 4 + {(\frac{3}{4})}^{7} \cdot 4}{4}$

Step 4.3

Raise $4$ to the power of $2$ .

$\frac{\sqrt{3}}{16} \cdot \frac{10 + \frac{9}{16} \cdot 4 + {(\frac{3}{4})}^{7} \cdot 4}{4}$

Step 4.4

Cancel the common factor of $4$ .

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Step 4.4.1

Factor $4$ out of $16$ .

$\frac{\sqrt{3}}{16} \cdot \frac{10 + \frac{9}{4 (4)} \cdot 4 + {(\frac{3}{4})}^{7} \cdot 4}{4}$

Step 4.4.2

Cancel the common factor.

$\frac{\sqrt{3}}{16} \cdot \frac{10 + \frac{9}{4 \cdot 4} \cdot 4 + {(\frac{3}{4})}^{7} \cdot 4}{4}$

Step 4.4.3

Rewrite the expression.

$\frac{\sqrt{3}}{16} \cdot \frac{10 + \frac{9}{4} + {(\frac{3}{4})}^{7} \cdot 4}{4}$

Step 4.5

Apply the product rule to $\frac{3}{4}$ .

$\frac{\sqrt{3}}{16} \cdot \frac{10 + \frac{9}{4} + \frac{3^{7}}{4^{7}} \cdot 4}{4}$

Step 4.6

Raise $3$ to the power of $7$ .

$\frac{\sqrt{3}}{16} \cdot \frac{10 + \frac{9}{4} + \frac{2187}{4^{7}} \cdot 4}{4}$

Step 4.7

Raise $4$ to the power of $7$ .

$\frac{\sqrt{3}}{16} \cdot \frac{10 + \frac{9}{4} + \frac{2187}{16384} \cdot 4}{4}$

Step 4.8

Cancel the common factor of $4$ .

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Step 4.8.1

Factor $4$ out of $16384$ .

$\frac{\sqrt{3}}{16} \cdot \frac{10 + \frac{9}{4} + \frac{2187}{4 (4096)} \cdot 4}{4}$

Step 4.8.2

Cancel the common factor.

$\frac{\sqrt{3}}{16} \cdot \frac{10 + \frac{9}{4} + \frac{2187}{4 \cdot 4096} \cdot 4}{4}$

Step 4.8.3

Rewrite the expression.

$\frac{\sqrt{3}}{16} \cdot \frac{10 + \frac{9}{4} + \frac{2187}{4096}}{4}$

Step 5

Find the common denominator.

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Step 5.1

Write $10$ as a fraction with denominator $1$ .

$\frac{\sqrt{3}}{16} \cdot \frac{\frac{10}{1} + \frac{9}{4} + \frac{2187}{4096}}{4}$

Step 5.2

Multiply $\frac{10}{1}$ by $\frac{4096}{4096}$ .

$\frac{\sqrt{3}}{16} \cdot \frac{\frac{10}{1} \cdot \frac{4096}{4096} + \frac{9}{4} + \frac{2187}{4096}}{4}$

Step 5.3

Multiply $\frac{10}{1}$ by $\frac{4096}{4096}$ .

$\frac{\sqrt{3}}{16} \cdot \frac{\frac{10 \cdot 4096}{4096} + \frac{9}{4} + \frac{2187}{4096}}{4}$

Step 5.4

Multiply $\frac{9}{4}$ by $\frac{1024}{1024}$ .

$\frac{\sqrt{3}}{16} \cdot \frac{\frac{10 \cdot 4096}{4096} + \frac{9}{4} \cdot \frac{1024}{1024} + \frac{2187}{4096}}{4}$

Step 5.5

Multiply $\frac{9}{4}$ by $\frac{1024}{1024}$ .

$\frac{\sqrt{3}}{16} \cdot \frac{\frac{10 \cdot 4096}{4096} + \frac{9 \cdot 1024}{4 \cdot 1024} + \frac{2187}{4096}}{4}$

Step 5.6

Multiply $4$ by $1024$ .

$\frac{\sqrt{3}}{16} \cdot \frac{\frac{10 \cdot 4096}{4096} + \frac{9 \cdot 1024}{4096} + \frac{2187}{4096}}{4}$

Step 6

Combine the numerators over the common denominator.

$\frac{\sqrt{3}}{16} \cdot \frac{\frac{10 \cdot 4096 + 9 \cdot 1024 + 2187}{4096}}{4}$

Step 7

Simplify each term.

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Step 7.1

Multiply $10$ by $4096$ .

$\frac{\sqrt{3}}{16} \cdot \frac{\frac{40960 + 9 \cdot 1024 + 2187}{4096}}{4}$

Step 7.2

Multiply $9$ by $1024$ .

$\frac{\sqrt{3}}{16} \cdot \frac{\frac{40960 + 9216 + 2187}{4096}}{4}$

Step 8

Simplify by adding numbers.

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Step 8.1

Add $40960$ and $9216$ .

$\frac{\sqrt{3}}{16} \cdot \frac{\frac{50176 + 2187}{4096}}{4}$

Step 8.2

Add $50176$ and $2187$ .

$\frac{\sqrt{3}}{16} \cdot \frac{\frac{52363}{4096}}{4}$

Step 9

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{3}}{16} \cdot (\frac{52363}{4096} \cdot \frac{1}{4})$

Step 10

Multiply $\frac{52363}{4096} \cdot \frac{1}{4}$ .

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Step 10.1

Multiply $\frac{52363}{4096}$ by $\frac{1}{4}$ .

$\frac{\sqrt{3}}{16} \cdot \frac{52363}{4096 \cdot 4}$

Step 10.2

Multiply $4096$ by $4$ .

$\frac{\sqrt{3}}{16} \cdot \frac{52363}{16384}$

Step 11

Multiply $\frac{\sqrt{3}}{16} \cdot \frac{52363}{16384}$ .

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Step 11.1

Multiply $\frac{\sqrt{3}}{16}$ by $\frac{52363}{16384}$ .

$\frac{\sqrt{3} \cdot 52363}{16 \cdot 16384}$

Step 11.2

Multiply $16$ by $16384$ .

$\frac{\sqrt{3} \cdot 52363}{262144}$

Step 12

Move $52363$ to the left of $\sqrt{3}$ .

$\frac{52363 \sqrt{3}}{262144}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{52363 \sqrt{3}}{262144}$

Decimal Form:

$0.34597540 \dots$