Mathematics is a wonderful language that deals with different types of numbers. Math is a broad subject that is divided in several different types of branches. In this section we are going to focus on the few important topics of algebra a pure branch of mathematics of grade VII. In this grade students usually study algebra 2. Algebra is nothing but it is simply the movement of the numbers, variables, and constants. If you love playing with numbers then definitely you we will love this subject. Now, come to the topic that is simplifying numerical expression.
Before we move to simplification of the numerical expression, let’s talk about the expression and simplification individually. Expression is a finite combination of the symbols that is well-defined and formed according to rules that depend upon the context. Symbols can designate numbers constant, variables, operators, functions, and other different types of mathematical symbols. These symbols includes punctuation, symbols of grouping, and other syntactic symbols. In algebra we generally deal with two different types of expression one is numeric and other is algebraic expression. An algebraic expression is that mathematical phrase that contains ordinary numbers, variables such as a, b ,x, y,.. and operators like add, subtract, multiply, and division. Let’s have a look on different types of algebraic expression.
a+ 2,
x- y,
6x,
c- 3/4d and other such type of expressions.
Now, the other type of expression that is called as the numeric expression, this means involving numbers and the term expression means “phrase”. In simple terms we can describe the same as: a mathematical phrase involving only numbers and one or more operational symbols.
Here are few examples of numerical expression,
5 + 20 – 7,
(1 + 3) – 7,
(6 × 2) ÷ 20,
4 ÷ (20 × 3),
7 × (42 + 3)
A numerical expression is the group of numerical or numeral values, which are separated by addition or subtraction. Numerical expression is simply the real number or all the positive values(1,2,3,4,5,6…) and negative values(-1,-2,-3,-4,-5,…) and zero. Order of process is one of the methods used to evaluate the numerical term in given expression.
Now, talk about simplification, the term simplification means reducing any expression to the least value for where we can divide the expression again. In other words we can say that simplification means solving the expression and removing all the complex or simple operators from the expression. Simplification of numerical expression means removing all the possible operators from the expression and taking the same to the simplest from, which will not contain any type of operator. The different numerical expression will always contain the numeric values only; They never contain the variables or any alphabetic value. While on the other side algebraic expressions contains both the numeric as well as variables with different type of operators.
Simplification of Numeric expression is very simply, if you knows how to perform different operations like addition, subtraction, multiplication, and divisions. All the operations are very simple and you have learned how to solve the problems of such type of operators in the early classes. This is the time when you have to implement that properly and solve different types of problems. In this article I will teach you how to simplify numeric expressions you can easily solve them by performing simple operations. Order of process is used when one or more function is included in any problem. We use three rules in order to deal with such kind of problems in order of operations.
> First one is perform the action within the parentheses.
> From left to right order, do all the multiplication and division function.
> Finally do all the addition and subtraction operations from left to right.
Let’s have few examples of the numerical expression and see how to apply above mentioned rules on the numerical expression.
Example: Simplify the numerical term in the expression 11+ 7 x (6 + 3) ÷ 9 - 8 using the order of operations.
Solution:
In this problem all the four symbols are present; we have to do all the four operations that includes: additions, subtraction, multiplication, division operations here.
= 11 + 7 x (6 +3) ÷ 9 – 8
=11+ 7 x 9 ÷ 9 – 8
= 11+ 63÷ 9 – 8
= 11 + 7 – 8
=18 – 8
= 10
Answer: 10
In this way we simplify the numerical expression. In the above expression we first solve the parentheses then on the priority basis we first perform the multiplication operation, then division after that addition and in last subtraction operation.
Example 2:
Solve the numerical expression 100÷ (13 + 3 x 9) - 2 using the order of operations.
Solution:
In this expression we are having all the four basic mathematical operators. So we have to perform all the four operations.
= 100 ÷ (13 + 3 x 9) – 2
=100 ÷ (13 + 27) – 2
= 100 ÷ 40 – 2
= 2.5– 2
= 0.5
Answer: 0.5
Example 3:
Get the answer of the given numerical expression 55 - (8 * 9 - 7) + 2 using the order of operations.
Solution: In the above mentioned expression we are having three operators and according to rule we first solve the parentheses then multiplication, after that addition and finally subtraction.
= 55 - (8 * 9 - 7) + 2
= 55 - (72 - 7) + 2
= 55 - 65 + 2
= 55 - 67
= -8
Answer: -8
Example 4:
Evaluate the numerical term in the expression 3 + 5 (2 + 4) - 7 using the order of operations.
Solution:
Since all the four symbols are present in the given expression, we have to do all the three additions, subtraction, multiplication, division operations here.
=5 + 5 ( 2 + 4 ) - 7
= 5+10+20 -7
= 35-7
= 28
Answer: 28
In this way we deal with the different type of numerical expression and simplify them to the simplest form. Whenever you deal with simplification of numerical expression always remember the order of operation rule and solve the problem using the same rule. If you don’t use the order of operation rule than the simple problem become the complex one and difficult to solve.
Now, talk about the other important topic that you study in grade VII i.e monomials.
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A monomial is that term which is comprised of a combination of the following: numbers, variables, and exponents. In algebraic expression and equations, terms, or monomials are separated by addition (+) and subtraction (-) symbols. In simple terms we can define a Monomial as an algebraic expression with only one term. For instance, 7ab, – 5p, 3z2, 4po, 5 etc. Monomial may have a constant or variables or both. To denote variables we use letters a,b,c ,d, x, y, l, m, ... etc. A variable can take various values; its value is not fixed. On the other side. a constant has a fixed value such as: 4, 500, 678,12432535, 100, –17, etc. Let’s put some light on the examples of Monomials |
1. 19xy
in this, Coefficient: 19 , variables x and y and exponent 1
2. -2ab2
in this expression Coefficient: -2 , variables are ‘a’ and ‘b’ and exponent is 2
3. 41 ab3
Coefficient: 41 , variables are a and b and exponent is 3
4. xa2
Coefficient: -1 because -x2 is the same as -1x2
Variable is x and the exponent is 2.
Let’s switch to the point that what operations can we perform on Monomials,
1. Addition and Subtraction of Monomials: When terms or monomials contain the same variable and same exponent, they are similar terms.
Addition and subtraction of monomials is done by combining the like terms. In addition we add the similar terms and in subtracting we subtract the similar or like terms.
Simplify the following expressions.
1) 7 + 7x +13x
In this simply add the x terms and leave the other terms as it is,
20x + 7.
2) -12c + 12c,
In this we are having both terms as the co-efficient of the c one co-efficient is -12 and other is 12. On simplifying the term we get 0.
3) 8y - 3y
Both the terms are of y on adding we get 5y.
4) x2 + y2 + x
we can’t simplify the above problem further as there is no like present in the Monomial.
Multiplication of Monomials: When you multiply the monomial, first step is multiplying the numerical coefficients (for e.g. 4 and the 8) and then multiplying the literal coefficients or variables (a and b). in the second step you need to multiply the similar variables by adding their exponents (for e.g. 3+2). (Rule am * an = am+n).
Simplify the following monomials:
1) 5 ab * 5 b = 25 ab2
2) 2 xy * 3 yz = 6 xy2z
3) -4 x6 * 6x2 = -24 x8
4) 4 b5c * 7 ab2c = 28 ab7c2
5) 20 ac * pq = 20acpq
Division of Monomials: When we perform this operation of the monomial the very first step is dividing the numerical coefficients (for e.g. 24 and 8) and then dividing the literal coefficients or variables (a and b). Second step is to divide the like variables by subtracting their exponents (for e. g. 5-2 ). (Rule am / an = am-n).
Simplify the following monomials:
1) 25 pq / 5 q = 5 p
2) 15 ab4c / 3 bc = 5 ab3
4) 49 x2y5z / 7 xy2z = 7 xy3
5) 20 ab / ab = 20
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