Basic Math Examples

$1 + \frac{1}{2^{n}} = \frac{5}{5120}$

Step 1

Cancel the common factor of $5$ and $5120$ .

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Step 1.1

Factor $5$ out of $5$ .

$1 + \frac{1}{2^{n}} = \frac{5 (1)}{5120}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $5$ out of $5120$ .

$1 + \frac{1}{2^{n}} = \frac{5 \cdot 1}{5 \cdot 1024}$

Step 1.2.2

Cancel the common factor.

$1 + \frac{1}{2^{n}} = \frac{5 \cdot 1}{5 \cdot 1024}$

Step 1.2.3

Rewrite the expression.

$1 + \frac{1}{2^{n}} = \frac{1}{1024}$

Step 2

Move all terms not containing $n$ to the right side of the equation.

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Step 2.1

Subtract $1$ from both sides of the equation.

$\frac{1}{2^{n}} = \frac{1}{1024} - 1$

Step 2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{1024}{1024}$ .

$\frac{1}{2^{n}} = \frac{1}{1024} - 1 \cdot \frac{1024}{1024}$

Step 2.3

Combine $- 1$ and $\frac{1024}{1024}$ .

$\frac{1}{2^{n}} = \frac{1}{1024} + \frac{- 1 \cdot 1024}{1024}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{1}{2^{n}} = \frac{1 - 1 \cdot 1024}{1024}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 1$ by $1024$ .

$\frac{1}{2^{n}} = \frac{1 - 1024}{1024}$

Step 2.5.2

Subtract $1024$ from $1$ .

$\frac{1}{2^{n}} = \frac{- 1023}{1024}$

Step 2.6

Move the negative in front of the fraction.

$\frac{1}{2^{n}} = - \frac{1023}{1024}$

Step 3

Multiply both sides by $2^{n}$ .

$\frac{1}{2^{n}} \cdot 2^{n} = - \frac{1023}{1024} \cdot 2^{n}$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $2^{n}$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{1}{2^{n}} \cdot 2^{n} = - \frac{1023}{1024} \cdot 2^{n}$

Step 4.1.1.2

Rewrite the expression.

$1 = - \frac{1023}{1024} \cdot 2^{n}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{1023}{1024} \cdot 2^{n}$ .

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Step 4.2.1.1

Combine $2^{n}$ and $\frac{1023}{1024}$ .

$1 = - \frac{2^{n} \cdot 1023}{1024}$

Step 4.2.1.2

Move $1023$ to the left of $2^{n}$ .

$1 = - \frac{1023 \cdot 2^{n}}{1024}$

Step 5

Solve for $n$ .

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Step 5.1

Rewrite the equation as $- \frac{1023 \cdot 2^{n}}{1024} = 1$ .

$- \frac{1023 \cdot 2^{n}}{1024} = 1$

Step 5.2

Multiply both sides of the equation by $- \frac{1024}{1023}$ .

$- \frac{1024}{1023} (- \frac{1023 \cdot 2^{n}}{1024}) = - \frac{1024}{1023} \cdot 1$

Step 5.3

Simplify both sides of the equation.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Simplify $- \frac{1024}{1023} (- \frac{1023 \cdot 2^{n}}{1024})$ .

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Step 5.3.1.1.1

Cancel the common factor of $1024$ .

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Step 5.3.1.1.1.1

Move the leading negative in $- \frac{1024}{1023}$ into the numerator.

$\frac{- 1024}{1023} (- \frac{1023 \cdot 2^{n}}{1024}) = - \frac{1024}{1023} \cdot 1$

Step 5.3.1.1.1.2

Move the leading negative in $- \frac{1023 \cdot 2^{n}}{1024}$ into the numerator.

$\frac{- 1024}{1023} \cdot \frac{- 1023 \cdot 2^{n}}{1024} = - \frac{1024}{1023} \cdot 1$

Step 5.3.1.1.1.3

Factor $1024$ out of $- 1024$ .

$\frac{1024 (- 1)}{1023} \cdot \frac{- 1023 \cdot 2^{n}}{1024} = - \frac{1024}{1023} \cdot 1$

Step 5.3.1.1.1.4

Cancel the common factor.

$\frac{1024 \cdot - 1}{1023} \cdot \frac{- 1023 \cdot 2^{n}}{1024} = - \frac{1024}{1023} \cdot 1$

Step 5.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{1023} (- 1023 \cdot 2^{n}) = - \frac{1024}{1023} \cdot 1$

Step 5.3.1.1.2

Cancel the common factor of $1023$ .

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Step 5.3.1.1.2.1

Factor $1023$ out of $- 1023 \cdot 2^{n}$ .

$\frac{- 1}{1023} (1023 (- 2^{n})) = - \frac{1024}{1023} \cdot 1$

Step 5.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{1023} (1023 (- 2^{n})) = - \frac{1024}{1023} \cdot 1$

Step 5.3.1.1.2.3

Rewrite the expression.

$- - 2^{n} = - \frac{1024}{1023} \cdot 1$

Step 5.3.1.1.3

Multiply.

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Step 5.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 \cdot 2^{n} = - \frac{1024}{1023} \cdot 1$

Step 5.3.1.1.3.2

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

$2^{n} = - \frac{1024}{1023} \cdot 1$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $- 1$ by $1$ .

$2^{n} = - \frac{1024}{1023}$

Step 5.4

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (2^{n}) = \ln (- \frac{1024}{1023})$

Step 5.5

The equation cannot be solved because $\ln (- \frac{1024}{1023})$ is undefined.

Undefined

Step 5.6

There is no solution for $2^{n} = - \frac{1024}{1023}$

No solution