Question

Solve the following:
$({x^4} + (1/{x^4}) \times (x + (1/x))$

Answer 1

When the equation has more than one algebraic operation, use the BODMAS rule. BODMAS rules says that firstly solve the brackets, then division, multiplication, addition and subtraction, in this specified manner itself or else there will be wrong answers. The above equation has many brackets, so we need to use the operations very carefully.

Complete answer: Equation in the question to:
$({x^4} + (1/{x^4}) \times (x + (1/x))$
To find: the final simplified equation
We need to solve the brackets first.
Let’s start with considering $({x^4} + \dfrac{1}{{{x^4}}}) = y$ ---1
When we replace the original equation with y, we get a simplified equation, i.e.
$y \times (x + \dfrac{1}{x})$
Multiplying y inside the bracket, we get,
$xy + \dfrac{y}{x}$
Now we substitute the value of y which we have obtained in equation 1, in the original equation.
After substituting, we get,
$x \times ({x^4} + \dfrac{1}{{{x^4}}}) + \dfrac{1}{x} \times ({x^4} + \dfrac{1}{{{x^4}}})$ ----2
Let us break the above equation in 2 parts.
We consider the first part as $x \times ({x^4} + \dfrac{1}{{{x^4}}})$
Multiplying x inside the bracket, we get,
${x^5} + \dfrac{x}{{{x^4}}} = {x^5} + \dfrac{1}{{{x^3}}}$ ---3
We consider the second part as $\dfrac{1}{x} \times ({x^4} + \dfrac{1}{{{x^4}}})$
Multiplying $\dfrac{1}{x}$ inside the bracket, we get,
$\dfrac{{{x^4}}}{x} + \dfrac{1}{{{x^5}}} = {x^3} + \dfrac{1}{{{x^5}}}$ ---4
Substituting equation 3 and equation 4 in equation 2, we get,
${x^5} + \dfrac{1}{{{x^3}}} + {x^3} + \dfrac{1}{{{x^5}}}$
Therefore after solving the given equation we get, ${x^5} + \dfrac{1}{{{x^3}}} + {x^3} + \dfrac{1}{{{x^5}}}$ as our final equation.

Note:
While solving such sums, BODMAS rules are very important to use. Brackets are given the 1st priority. In this sum, the value of any unknown variable is not given, therefore there won’t be any numerical answer obtained. Your final answer will also be an equation in such cases. Power or degree should be calculated properly after multiplication or division. Also, variables with different power cannot be added or subtracted.