In algebra, real numbers have a number of properties that enable us to simplify and solve mathematical problems. Its density property for example, implies that there is always another real number that exists between any two real numbers given. For example, between 9.1 and 9.2, there are 9.11, 9.12, 9.13 and so forth. Its identity property says that any number added to zero is equal to itself. There are a number of other properties that real numbers have including distributive, commutative, reflexive and symmetric among others. They all serve the same purpose of helping us in problem solving.

For easier computation it is always important to cluster like terms together. In more complex expressions, the like terms must have the same variable raised to the same exponent.

During the simplification of the given expressions, properties of real numbers will be utilized and identified. The mathematical workings will be aligned on the left while the right side will discuss the properties used.

2a(a-5)+4(a-5) The given expression

2a2-10a+4a-20 The parentheses are removed by use of the distributive property

2a2-6a-20 Coefficients are added to enable combination of like terms

The above is simplified fully and no more computation is needed. The like terms were already grouped together and as such there was no need to rearrange the order.

2w-3+3(w-4)-5(w-6) The given expression..

2w-3+3w-12-5w+30 Parentheses removed by distributive property.

2w+3w-5w-12+30 Commutative property of real numbers is used to arrange the like terms together. -12 and 30 are like constant terms while 2w, 3w and -5w are like variable terms. These can then all be added or subtracted.

5w-5w+18Two of both variable terms and constant terms are added.

18The remaining pair of like variable terms is added.

The expression above is fully simplified. The variable terms cancel each other out through subtraction and a single constant term remains as the simplified form.

0.05(0.3m+35n)-0.8(-0.09n-22m)The given expression

0.015m+1.75n+0.072n+17.6m Removal of parentheses using the distributive property.

0.015m+17.6m+1.75n+0.072nThe commutative property enables us arrange like terms together. Only variable terms with the same variable may be combined

17.615m+1.822n Addition of coefficients enable combination of like terms

This given problem looked complicated but it is not substantially different from the others as it simply employs the usage of the decimal numbers.

From the three expressions above it is clear that knowing the properties of real numbers is important. It is also important to arrange mathematical workings in a chronological and organized manner to enable faster understanding and conceptualization of how to tackle different mathematical problems that are encountered.


Poole, D. (2014). Linear algebra: A modern introduction. Cengage Learning.

Samuel, P. (2013). Algebraic Theory of Numbers: Translated from the French by Allan J. Silberger. Courier Dover Publications.