Basic Math Examples

$K = \frac{1}{\frac{1}{K} + \frac{1}{K^{2}}}$

Step 1

Factor each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

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

$K = \frac{1}{\frac{1}{K} \cdot \frac{K}{K} + \frac{1}{K^{2}}}$

Step 1.1.2

Write each expression with a common denominator of $K^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.2.1

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

$K = \frac{1}{\frac{K}{K \cdot K} + \frac{1}{K^{2}}}$

Step 1.1.2.2

Multiply $K$ by $K$ .

$K = \frac{1}{\frac{K}{K^{2}} + \frac{1}{K^{2}}}$

Step 1.1.3

Combine the numerators over the common denominator.

$K = \frac{1}{\frac{K + 1}{K^{2}}}$

Step 1.2

Multiply the numerator by the reciprocal of the denominator.

$K = 1 \frac{K^{2}}{K + 1}$

Step 1.3

Multiply $\frac{K^{2}}{K + 1}$ by $1$ .

$K = \frac{K^{2}}{K + 1}$

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.

$1, K + 1$

Step 2.2

Remove parentheses.

$1, K + 1$

Step 2.3

The LCM of one and any expression is the expression.

$K + 1$

Step 3

Multiply each term in $K = \frac{K^{2}}{K + 1}$ by $K + 1$ to eliminate the fractions.

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Step 3.1

Multiply each term in $K = \frac{K^{2}}{K + 1}$ by $K + 1$ .

$K (K + 1) = \frac{K^{2}}{K + 1} (K + 1)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

$K \cdot K + K \cdot 1 = \frac{K^{2}}{K + 1} (K + 1)$

Step 3.2.2

Simplify the expression.

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Step 3.2.2.1

Multiply $K$ by $K$ .

$K^{2} + K \cdot 1 = \frac{K^{2}}{K + 1} (K + 1)$

Step 3.2.2.2

Multiply $K$ by $1$ .

$K^{2} + K = \frac{K^{2}}{K + 1} (K + 1)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $K + 1$ .

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Step 3.3.1.1

Cancel the common factor.

$K^{2} + K = \frac{K^{2}}{K + 1} (K + 1)$

Step 3.3.1.2

Rewrite the expression.

$K^{2} + K = K^{2}$

Step 4

Move all terms containing $K$ to the left side of the equation.

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Step 4.1

Subtract $K^{2}$ from both sides of the equation.

$K^{2} + K - K^{2} = 0$

Step 4.2

Combine the opposite terms in $K^{2} + K - K^{2}$ .

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Step 4.2.1

Subtract $K^{2}$ from $K^{2}$ .

$K + 0 = 0$

Step 4.2.2

Add $K$ and $0$ .

$K = 0$