Finite Math Examples

${(\frac{\sqrt{3}}{2})}^{14} - {(\frac{1}{2} \cdot i)}^{14}$

Step 1

Simplify each term.

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Step 1.1

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

$\frac{{\sqrt{3}}^{14}}{2^{14}} - {(\frac{1}{2} \cdot i)}^{14}$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Rewrite ${\sqrt{3}}^{14}$ as $3^{7}$ .

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Step 1.2.1.1

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

$\frac{{(3^{\frac{1}{2}})}^{14}}{2^{14}} - {(\frac{1}{2} \cdot i)}^{14}$

Step 1.2.1.2

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

$\frac{3^{\frac{1}{2} \cdot 14}}{2^{14}} - {(\frac{1}{2} \cdot i)}^{14}$

Step 1.2.1.3

Combine $\frac{1}{2}$ and $14$ .

$\frac{3^{\frac{14}{2}}}{2^{14}} - {(\frac{1}{2} \cdot i)}^{14}$

Step 1.2.1.4

Cancel the common factor of $14$ and $2$ .

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Step 1.2.1.4.1

Factor $2$ out of $14$ .

$\frac{3^{\frac{2 \cdot 7}{2}}}{2^{14}} - {(\frac{1}{2} \cdot i)}^{14}$

Step 1.2.1.4.2

Cancel the common factors.

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Step 1.2.1.4.2.1

Factor $2$ out of $2$ .

$\frac{3^{\frac{2 \cdot 7}{2 (1)}}}{2^{14}} - {(\frac{1}{2} \cdot i)}^{14}$

Step 1.2.1.4.2.2

Cancel the common factor.

$\frac{3^{\frac{2 \cdot 7}{2 \cdot 1}}}{2^{14}} - {(\frac{1}{2} \cdot i)}^{14}$

Step 1.2.1.4.2.3

Rewrite the expression.

$\frac{3^{\frac{7}{1}}}{2^{14}} - {(\frac{1}{2} \cdot i)}^{14}$

Step 1.2.1.4.2.4

Divide $7$ by $1$ .

$\frac{3^{7}}{2^{14}} - {(\frac{1}{2} \cdot i)}^{14}$

Step 1.2.2

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

$\frac{2187}{2^{14}} - {(\frac{1}{2} \cdot i)}^{14}$

Step 1.3

Raise $2$ to the power of $14$ .

$\frac{2187}{16384} - {(\frac{1}{2} \cdot i)}^{14}$

Step 1.4

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

$\frac{2187}{16384} - {(\frac{i}{2})}^{14}$

Step 1.5

Apply the product rule to $\frac{i}{2}$ .

$\frac{2187}{16384} - \frac{i^{14}}{2^{14}}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Rewrite $i^{14}$ as ${(i^{4})}^{3} i^{2}$ .

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Step 1.6.1.1

Factor out $i^{12}$ .

$\frac{2187}{16384} - \frac{i^{12} i^{2}}{2^{14}}$

Step 1.6.1.2

Rewrite $i^{12}$ as ${(i^{4})}^{3}$ .

$\frac{2187}{16384} - \frac{{(i^{4})}^{3} i^{2}}{2^{14}}$

Step 1.6.2

Rewrite $i^{4}$ as $1$ .

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Step 1.6.2.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$\frac{2187}{16384} - \frac{{({(i^{2})}^{2})}^{3} i^{2}}{2^{14}}$

Step 1.6.2.2

Rewrite $i^{2}$ as $- 1$ .

$\frac{2187}{16384} - \frac{{({(- 1)}^{2})}^{3} i^{2}}{2^{14}}$

Step 1.6.2.3

Raise $- 1$ to the power of $2$ .

$\frac{2187}{16384} - \frac{1^{3} i^{2}}{2^{14}}$

Step 1.6.3

One to any power is one.

$\frac{2187}{16384} - \frac{1 i^{2}}{2^{14}}$

Step 1.6.4

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

$\frac{2187}{16384} - \frac{i^{2}}{2^{14}}$

Step 1.6.5

Rewrite $i^{2}$ as $- 1$ .

$\frac{2187}{16384} - \frac{- 1}{2^{14}}$

Step 1.7

Raise $2$ to the power of $14$ .

$\frac{2187}{16384} - \frac{- 1}{16384}$

Step 1.8

Move the negative in front of the fraction.

$\frac{2187}{16384} - - \frac{1}{16384}$

Step 1.9

Multiply $- - \frac{1}{16384}$ .

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Step 1.9.1

Multiply $- 1$ by $- 1$ .

$\frac{2187}{16384} + 1 (\frac{1}{16384})$

Step 1.9.2

Multiply $\frac{1}{16384}$ by $1$ .

$\frac{2187}{16384} + \frac{1}{16384}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{2187 + 1}{16384}$

Step 2.2

Add $2187$ and $1$ .

$\frac{2188}{16384}$

Step 2.3

Cancel the common factor of $2188$ and $16384$ .

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Step 2.3.1

Factor $4$ out of $2188$ .

$\frac{4 (547)}{16384}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $4$ out of $16384$ .

$\frac{4 \cdot 547}{4 \cdot 4096}$

Step 2.3.2.2

Cancel the common factor.

$\frac{4 \cdot 547}{4 \cdot 4096}$

Step 2.3.2.3

Rewrite the expression.

$\frac{547}{4096}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{547}{4096}$

Decimal Form:

$0.13354492 \dots$