Question

How do you simplify$\dfrac{1}{x} + \dfrac{1}{{{x^2}}}$?

Answer 1

Hint:Addition of the form $\dfrac{a}{b} + \dfrac{c}{d}$ is done as $\dfrac{{\left[ {a \times \dfrac{{lcm(b,d)}}{b}} \right] + \left[ {c \times \dfrac{{lcm(b,d)}}{d}} \right]}}{{lcm(b,d)}}$
To solve these types of fractions, the first step we need to do is to find the LCM of the denominators. After completing this step, you will have to divide the LCM with the opposite denominator and multiply the answer with the numerator of the other fraction. This step mathematically looks like $\dfrac{{\left[ {a \times \dfrac{{lcm(b,d)}}{b}} \right] + \left[ {c \times \dfrac{{lcm(b,d)}}{d}} \right]}}{{lcm(b,d)}}$

Complete step by step answer:
The given algebraic expression we have is $\dfrac{1}{x} + \dfrac{1}{{{x^2}}}$.To simplify this, the first step we will be doing is to add these two. Now, we know that whenever two numbers of the form $\dfrac{a}{b}$and $\dfrac{c}{d}$ are added or subtracted we get the answer of the form,
$\dfrac{{\left[ {a \times \dfrac{{lcm(b,d)}}{b}} \right] \pm \left[ {c \times \dfrac{{lcm(b,d)}}{d}} \right]}}{{lcm(b,d)}}$
So, when we compare our expression with this we get that here,
$d = {x^2}$,$b = {x^2}$, $c = 1$and $d = {x^2}$
Therefore, if we use the above formula we get the answer as
\[\dfrac{{\left[ {\dfrac{{{x^2}}}{x}} \right] + \left[ {\dfrac{{{x^2}}}{{{x^2}}}} \right]}}{{{x^2}}} = \dfrac{{x + 1}}{{{x^2}}}\]
Cancelling out the common terms, we get
\[\dfrac{{\left[ {\dfrac{{{x^2}}}{x}} \right] + \left[ {\dfrac{{{x^2}}}{{{x^2}}}} \right]}}{{{x^2}}} = \dfrac{{x + 1}}{{{x^2}}}\]

Hence, the simplified form of the algebraic form $\dfrac{1}{x} + \dfrac{1}{{{x^2}}}$is \[\dfrac{{x + 1}}{{{x^2}}}\].

Note:The formula to add two fractions only looks big but isn’t a big task to remember. Only after solving 5-6 problems will you get to know how to do it. All you have to do is find the LCM of both the numbers (denominators) and then add the respective values in the numerator shown in the formula.Another point to remember is, whenever you see two prime numbers in the denominator. Just multiply those two numbers to get the LCM. This method can also be used to find the LCM of two prime numbers. Because for prime numbers, the LCM is the multiplication of the two numbers itself, as there are no other factors.