The multiplicative inverse of a number is understood as another number that multiplied by the first gives the neutral element of the product, that is, the unit. If we have a real number a then its multiplicative inverse is denoted by a -1, and it is true that:
aa -1 = a -1 a = 1
In general, the number a belongs to the set of real numbers.
Figure 1. Y is the multiplicative inverse of X and X is the multiplicative inverse of Y.
If for example we take a = 2, then its multiplicative inverse is 2 -1 = ½ since the following holds:
2 ⋅ 2 -1 = 2 -1 ⋅ 2 = 1
2⋅ ½ = ½ ⋅ 2 = 1
The multiplicative inverse of a number is also called the reciprocal, because the multiplicative inverse is obtained by exchanging numerator and denominator, for example the multiplicative inverse of 3/4 is 4/3.
As a general rule it can be said that for a rational number (p / q) its multiplicative inverse (p / q) -1 is reciprocal (q / p) as can be verified below:
(p / q) ⋅ (p / q) -1 = (p / q) ⋅ (q / p) = (p⋅ q) / (q⋅ p) = (p⋅ q) / (p⋅ q) = one
Recall that the multiplicative inverse is also called the reciprocal because it is obtained precisely by exchanging numerator and denominator.
Then the multiplicative inverse of (a - b) / (a ^ 2 - b ^ 2) will be:
(a ^ 2 - b ^ 2) / (a - b)
But this expression can be simplified if we recognize, according to the rules of algebra, that the numerator is a difference of squares that can be factored as the product of a sum by a difference:
((a + b) (a - b)) / (a - b)
As there is a common factor (a - b) in the numerator and in the denominator, we proceed to simplify, finally obtaining:
(a + b) which is the multiplicative inverse of (a - b) / (a ^ 2 - b ^ 2).
References
- Fuentes, A. (2016). BASIC MATH. An Introduction to Calculus. Lulu.com.
- Garo, M. (2014). Mathematics: quadratic equations: How solve a quadratic equation. Marilù Garo.
- Haeussler, EF, & Paul, RS (2003). Mathematics for management and economics. Pearson Education.
- Jiménez, J., Rofríguez, M., & Estrada, R. (2005). Math 1 SEP. Threshold.
- Preciado, CT (2005). Mathematics Course 3rd. Editorial Progreso.
- Rock, NM (2006). Algebra I Is Easy! So Easy. Team Rock Press.
- Sullivan, J. (2006). Algebra and Trigonometry. Pearson Education.