What is the dash formula?

The dash operator, sometimes referred to as the dash formula or the shortcut formula, offers a condensed approach for calculating the sum or difference of two ratios with comparable numerators. This strategy is helpful in simplifying rational expressions, particularly when the denominators are cumbersome or include challenging elements like square roots or radical expressions.

The dash formula is defined as follows:

(a + b)/(c + d) - (a - b)/(c - d) = (4ab)/(c^2 - d^2)

where:

a and b are the numerators of the two fractions

c and d are the denominators of the two fractions

Understanding the dash formula requires analyzing its components. The left side of the formula comprises the difference of two fractions, while the right side demonstrates how to express this difference in a simplified form using solely the numerators and denominators of the original fractions, along with the operation of multiplication.

Here are the steps to use the dash formula:

Find the product of the numerators: Multiply the numerator of the first fraction (a + b) by the numerator of the second fraction (a - b).

Find the product of the denominators: Multiply the denominator of the first fraction (c + d) by the denominator of the second fraction (c - d).

Subtract the product of the numerators from the product of the denominators: This difference forms the numerator of the simplified fraction.

Place the result over the square of the difference between the denominators: This becomes the denominator of the simplified fraction.

The end outcome of employing the dash formula is a simplified fraction with the same value as the original difference of the two fractions. This strategy proves particularly beneficial in situations where the simplification of complex rational expressions is required, especially in algebraic calculations or mathematical proofs.