Chapter 5.6, Problem 20WE

To determine

To simplify the given expression.

Answer to Problem 20WE

  $\frac{4y^2+4xy+x^2}{4x^2{y}^2}$

Explanation of Solution

Given:

Expression: $\frac{1}{x^2}+\frac{1}{xy}+\frac{1}{4y^2}$

Calculation:

Consider the given expression $\frac{1}{x^2}+\frac{1}{xy}+\frac{1}{4y^2}$ ,

Since the denominators of both the fractions are different, find the least common denominator for both the fractions.

Here, least common denominator is $4x^2{y}^2$ , so

  $\frac{1}{x^2}+\frac{1}{xy}+\frac{1}{4y^2}$

  $=\frac{1\cdot 4y^2}{x^2\cdot 4y^2}+\frac{1\cdot 4xy}{xy\cdot 4xy}+\frac{1\cdot x^2}{4y^2\cdot x^2}$

  $=\frac{4y^2}{4x^2{y}^2}+\frac{4xy}{4x^2{y}^2}+\frac{x^2}{4x^2{y}^2}$

Now, as the denominators of the two fractions are same, simply add the numerators and put the result on the same denominator.

  $=\frac{4y^2+4xy+x^2}{4x^2{y}^2}$

Conclusion:

Therefore, after simplification, the answer is $\frac{4y^2+4xy+x^2}{4x^2{y}^2}$ .