Solve for R 1/R=1/R_1+1/R_2

$\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}}$

Step 1

Find the LCD of the terms in the equation.

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

R,B,C

Since $R, R_{1}, R_{2}$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1$ then find LCM for the variable part $R^{1}, {R_{1}}^{1}, {R_{2}}^{1}$ .

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

The factor for $R^{1}$ is $R$ itself.

$R^{1} = R$

$R$ occurs $1$ time.

The factor for ${R_{1}}^{1}$ is $R_{1}$ itself.

${R_{1}}^{1} = R_{1}$

$R_{1}$ occurs $1$ time.

The factor for ${R_{2}}^{1}$ is $R_{2}$ itself.

${R_{2}}^{1} = R_{2}$

$R_{2}$ occurs $1$ time.

The LCM of $R^{1}, {R_{1}}^{1}, {R_{2}}^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$R \cdot R_{1} \cdot R_{2}$

Multiply $R \cdot R_{1}$ by $R_{2}$ .

$R \cdot R_{1} R_{2}$

Step 2

Multiply each term in $\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}}$ by $R \cdot R_{1} R_{2}$ to eliminate the fractions.

Multiply each term in $\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}}$ by $R \cdot R_{1} R_{2}$ .

$\frac{1}{R} (R \cdot R_{1} R_{2}) = \frac{1}{R_{1}} (R \cdot R_{1} R_{2}) + \frac{1}{R_{2}} (R \cdot R_{1} R_{2})$

Simplify the left side.

Cancel the common factor of $R$ .

Factor $R$ out of $R \cdot R_{1} R_{2}$ .

$\frac{1}{R} (R (R_{1} R_{2})) = \frac{1}{R_{1}} (R \cdot R_{1} R_{2}) + \frac{1}{R_{2}} (R \cdot R_{1} R_{2})$

Cancel the common factor.

$\frac{1}{R} (R (R_{1} R_{2})) = \frac{1}{R_{1}} (R \cdot R_{1} R_{2}) + \frac{1}{R_{2}} (R \cdot R_{1} R_{2})$

Rewrite the expression.

$R_{1} R_{2} = \frac{1}{R_{1}} (R \cdot R_{1} R_{2}) + \frac{1}{R_{2}} (R \cdot R_{1} R_{2})$

Simplify the right side.

Simplify each term.

Cancel the common factor of $R_{1}$ .

Factor $R_{1}$ out of $R \cdot R_{1} R_{2}$ .

$R_{1} (R_{2}) = \frac{1}{R_{1}} (R_{1} (R (R_{2}))) + \frac{1}{R_{2}} (R (R_{1}) R_{2})$

Cancel the common factor.

$R_{1} (R_{2}) = \frac{1}{R_{1}} (R_{1} (R (R_{2}))) + \frac{1}{R_{2}} (R (R_{1}) R_{2})$

Rewrite the expression.

$R_{1} R_{2} = R \cdot R_{2} + \frac{1}{R_{2}} (R \cdot R_{1} R_{2})$

Cancel the common factor of $R_{2}$ .

Factor $R_{2}$ out of $R \cdot R_{1} R_{2}$ .

$R_{1} (R_{2}) = R (R_{2}) + \frac{1}{R_{2}} (R_{2} (R \cdot R_{1}))$

Cancel the common factor.

$R_{1} (R_{2}) = R (R_{2}) + \frac{1}{R_{2}} (R_{2} (R \cdot R_{1}))$

Rewrite the expression.

$R_{1} R_{2} = R \cdot R_{2} + R \cdot R_{1}$

Step 3

Solve the equation.

Rewrite the equation as $R \cdot R_{2} + R \cdot R_{1} = R_{1} R_{2}$ .

$R \cdot R_{2} + R \cdot R_{1} = R_{1} R_{2}$

Factor $R$ out of $R \cdot R_{2} + R \cdot R_{1}$ .

Factor $R$ out of $R \cdot R_{2}$ .

$R (R_{2}) + R \cdot R_{1} = R_{1} (R_{2})$

Factor $R$ out of $R \cdot R_{1}$ .

$R (R_{2}) + R (R_{1}) = R_{1} (R_{2})$

Factor $R$ out of $R (R_{2}) + R (R_{1})$ .

$R (R_{2} + R_{1}) = R_{1} R_{2}$

Divide each term in $R (R_{2} + R_{1}) = R_{1} R_{2}$ by $R_{2} + R_{1}$ and simplify.

Divide each term in $R (R_{2} + R_{1}) = R_{1} R_{2}$ by $R_{2} + R_{1}$ .

$\frac{R (R_{2} + R_{1})}{R_{2} + R_{1}} = \frac{R_{1} R_{2}}{R_{2} + R_{1}}$

Simplify the left side.

Cancel the common factor of $R_{2} + R_{1}$ .

Cancel the common factor.

$\frac{R (R_{2} + R_{1})}{R_{2} + R_{1}} = \frac{R_{1} R_{2}}{R_{2} + R_{1}}$

Divide $R$ by $1$ .

$R = \frac{R_{1} R_{2}}{R_{2} + R_{1}}$

Solve for R 1/R=1/R_1+1/R_2

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