Numbers in the form of x/y where y = 0 can be called Fractions. A fraction represents part of a whole quantity, in other words, a fraction denotes part of a complete entity. Here in x/y, x denotes the numerator of the fraction and y denotes the denominator of the fraction. Both x and y can be made up of constants, variables, decimals, exponents, etc. Fractions can be both positive or negative and are called Positive Fractions and Negative Fractions depending on the sign of the operator with them.

Fractions can further be categorized into 2 categories: 

1. Proper Fraction: Here, the numerator is smaller than the denominator. Given by x < y, where x/y is the fraction. This is the most desired form of a fraction.
2. Improper Fraction: Here, the numerator is greater than the denominator. Given by x > y, where x/y is the fraction. This is not a very desired form of a fraction. Improper fractions are often denoted as Mixed Fractions.

Simplifying fractions

For simplifying fractions the like constants, like variables, like decimals, and likewise like exponents are clubbed separately and then mathematical operations are performed on the basis of the feasibility of the operation. Like variables are directly added/subtracted/multiplied/divided.

Similarly, in the approach for exponent also first like exponents are combined for performing mathematical operations. 

Steps for simplifying fractions with variables and exponents 

Step 1: Check for like terms, be it for variables or exponents, and place them close to each other. 

Step 2: If the exponents are small then replace them with the value of the exponent.

Step 3: For simplifying the expression, perform mathematical operations as desired on the Like Terms placed together.

Step 4: The combined terms, exponents, unlike terms all of them simplified, are placed together in the final step of the solution.

Sample Problems

Question 1: Simplify the expression: 1x/2 + 3x/2 +6² – 5

Solution: 

Combining the terms with the variable x

= 1x/2 + 3x/2 + 6² – 5

= 4x/2 + 6² – 5

= 2x + 36 -5

= 2x + 31

Question 2: Simplify the expression: x/2 + 3y/2 +6² – e²

Solution:

Combining of the terms can’t take place as we have two different variables x and y

= x/2 + 3y/2 + 6² – e²

= x/2 + 3y/2 + 6² – e²

= x/2 + 3y/2 + 36 – e²

Question 3: Simplify the expression: x/2 + 3/2 +6² – e²

Solution:

= x/2 + 3/2 + 6² – e²

= x/2 + 3/2 + 6² – e²

= x/2 + 3/2 + 36 – e²

= x/2 + 3/2 + 36 × 2/2 – e²

= x/2 + 75/2 – e²

Question 4: Simplify the expression: z/2 + 3/2 +2z – e⁵

Solution:

Combining terms with variable ‘z’

= z/2 + 3/2 + 2z – e⁵

= z/2 + 2z + 3/2  – e⁵

= z/2 + (2z × 2)/(1 × 2) + 3/2  – e⁵

= z/2 + 4z/2 + 3/2  – e⁵

Question 5: Simplify the expression: x/4 + y/5 + 3x/4  – e^-4 + 3⁴

Solution:

Combining terms with Like Variables

x/4 + 3x/4 + y/5 – e^-4 + 3⁴

The expression now becomes:

4x/4 + y/5 – e^-4 + 81

x + y/5 – e^-4 + 81

Question 6: Simplify the expression: z + 3x/5 +6² – 5²

Solution:

Since the variables are different, they can’t be combined.

= z + 3x/5 + 6² – 5²

= z + 3x/5 + 36 – 25

= z + 3x/5 +11