Finite Math Examples

$n = \frac{i n^{0.1}}{i n^{0.933}}$

Step 1

Factor each term.

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Step 1.1

Reduce the expression $\frac{i n^{0.1}}{i n^{0.933}}$ by cancelling the common factors.

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Step 1.1.1

Cancel the common factor.

$n = \frac{i n^{0.1}}{i n^{0.933}}$

Step 1.1.2

Rewrite the expression.

$n = \frac{n^{0.1}}{n^{0.933}}$

Step 1.2

Move $n^{0.1}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$n = \frac{1}{n^{0.933} n^{- 0.1}}$

Step 1.3

Multiply $n^{0.933}$ by $n^{- 0.1}$ by adding the exponents.

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Step 1.3.1

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

$n = \frac{1}{n^{0.933 - 0.1}}$

Step 1.3.2

Subtract $0.1$ from $0.933$ .

$n = \frac{1}{n^{0.833}}$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, n^{0.833}$

Step 2.2

The LCM of one and any expression is the expression.

$n^{0.833}$

Step 3

Multiply each term in $n = \frac{1}{n^{0.833}}$ by $n^{0.833}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $n = \frac{1}{n^{0.833}}$ by $n^{0.833}$ .

$n \cdot n^{0.833} = \frac{1}{n^{0.833}} n^{0.833}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Multiply $n$ by $n^{0.833}$ by adding the exponents.

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Step 3.2.1.1

Multiply $n$ by $n^{0.833}$ .

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Step 3.2.1.1.1

Raise $n$ to the power of $1$ .

$n^{1} n^{0.833} = \frac{1}{n^{0.833}} n^{0.833}$

Step 3.2.1.1.2

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

$n^{1 + 0.833} = \frac{1}{n^{0.833}} n^{0.833}$

Step 3.2.1.2

Add $1$ and $0.833$ .

$n^{1.833} = \frac{1}{n^{0.833}} n^{0.833}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $n^{0.833}$ .

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Step 3.3.1.1

Cancel the common factor.

$n^{1.833} = \frac{1}{n^{0.833}} n^{0.833}$

Step 3.3.1.2

Rewrite the expression.

$n^{1.833} = 1$

Step 4

Solve the equation.

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Step 4.1

Convert the decimal exponent to a fractional exponent.

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Step 4.1.1

Convert the decimal number to a fraction by placing the decimal number over a power of ten. Since there are $3$ numbers to the right of the decimal point, place the decimal number over $10^{3}$ $(1000)$ . Next, add the whole number to the left of the decimal.

$n^{1 \frac{833}{1000}} = 1$

Step 4.1.2

Convert $1 \frac{833}{1000}$ to an improper fraction.

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Step 4.1.2.1

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

$n^{1 + \frac{833}{1000}} = 1$

Step 4.1.2.2

Add $1$ and $\frac{833}{1000}$ .

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Step 4.1.2.2.1

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

$n^{\frac{1000}{1000} + \frac{833}{1000}} = 1$

Step 4.1.2.2.2

Combine the numerators over the common denominator.

$n^{\frac{1000 + 833}{1000}} = 1$

Step 4.1.2.2.3

Add $1000$ and $833$ .

$n^{\frac{1833}{1000}} = 1$

Step 4.2

Raise each side of the equation to the power of $\frac{1}{1.833}$ to eliminate the fractional exponent on the left side.

${(n^{\frac{1833}{1000}})}^{\frac{1}{1.833}} = 1^{\frac{1}{1.833}}$

Step 4.3

Simplify the exponent.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Simplify ${(n^{\frac{1833}{1000}})}^{\frac{1}{1.833}}$ .

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Step 4.3.1.1.1

Multiply the exponents in ${(n^{\frac{1833}{1000}})}^{\frac{1}{1.833}}$ .

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Step 4.3.1.1.1.1

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

$n^{\frac{1833}{1000} \cdot \frac{1}{1.833}} = 1^{\frac{1}{1.833}}$

Step 4.3.1.1.1.2

Cancel the common factor of $1.833$ .

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Step 4.3.1.1.1.2.1

Factor $1.833$ out of $1833$ .

$n^{\frac{1.833 (1000)}{1000} \cdot \frac{1}{1.833}} = 1^{\frac{1}{1.833}}$

Step 4.3.1.1.1.2.2

Cancel the common factor.

$n^{\frac{1.833 \cdot 1000}{1000} \cdot \frac{1}{1.833}} = 1^{\frac{1}{1.833}}$

Step 4.3.1.1.1.2.3

Rewrite the expression.

$n^{\frac{1000}{1000}} = 1^{\frac{1}{1.833}}$

Step 4.3.1.1.1.3

Divide $1000$ by $1000$ .

$n^{1} = 1^{\frac{1}{1.833}}$

Step 4.3.1.1.2

Simplify.

$n = 1^{\frac{1}{1.833}}$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Simplify $1^{\frac{1}{1.833}}$ .

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Step 4.3.2.1.1

Divide $1$ by $1.833$ .

$n = 1^{0.54555373}$

Step 4.3.2.1.2

One to any power is one.

$n = 1$

Step 5

Exclude the solutions that do not make $n = \frac{i n^{0.1}}{i n^{0.933}}$ true.

$No solution$