Value of an Expression: Definitions, Examples, Problems

Value of an Expression: Expressions are mathematical statements with at least one term containing numbers or variables, or both, linked by an operator. Addition, subtraction, multiplication, and division are examples of mathematical operators. For instance, \(x+y\) is an expression in which \(x\) and \(y\) are terms separated by an addition operator. Arithmetic expressions, which contain simply numbers, and algebraic expressions, which include both numbers and variables, are the two forms of expressions in mathematics. Mathematical expressions solve complicated problems by finding their value.

When the variables and constants of a mathematical expression are assigned values, the result of the computation described by this expression is the value. In this article, let’s learn everything about of value of an expression. Read on to find more.

Expressions

In mathematics, an expression is a statement that has at least two numbers and one math operation. Let’s have a look at how to write expressions.

Example: A number is \(8\), more than half the other number, and the different number is \(x\).

This statement is written as \(\frac{x}{2}+8\) in a mathematical expression.

Arithmetic or Numerical Expressions

The arithmetic/numerical expression is a mathematical statement consisting of numbers and one or more operation symbols. Addition, subtraction, multiplication, and division are examples of operation symbols.

Arithmetic/numerical expressions are composed of integers, operators, parentheses, and variables syntactically accurate.

Example: \(\left(10^{2}-30\right)+50\).

\(\Rightarrow(100-30)+50\)

\(\Rightarrow 70+50\)

\(\Rightarrow 120\)

Learn the Concepts of Algebraic Expressions

Value of Arithmetic/Numerical Expressions

The value of a numerical expression is the value obtained by solving the arithmetic/numerical expression.

We understand how to perform the four basic operations on whole numbers, fractions, and decimals: addition, subtraction, multiplication, and division. We only perform one operation at a time. Now we will look at how to combine two or more operations.

To simplify and obtain the value of a numerical statement with two or more operations, we perform operations such as division first, followed by multiplication, addition, and subtraction. These operations are carried out using a standard result known as BODMAS.

The word BODMAS stands for:

\(B →\) Brackets

\(O →\) Order or Exponents

\(D →\) Division

\(M →\) Multiplication

\(A →\) Addition

\(S →\) Subtraction

If the brackets are present in the problem, we first simplify the brackets. There are four kinds of brackets.

1. \(() \rightarrow\) simple brackets or round brackets or parenthesis.
   
2. \(\{ \} \rightarrow\) Braces or curly brackets.
   
3. \([\,] \rightarrow\) Square brackets.

4. _____________\(\rightarrow\) This is a line called a bar, vinculum. It is placed either overline or as an underline.
   If two or more types of brackets are involved in the problem, then they are removed in this order ‘_______________’, \((),~\{ \} ,~[\,]\).

Examples:

1. \(\left[ {13 + \left\{ {7 – \left( {8 \div 2} \right)} \right\}} \right] \times 3\)

\( \Rightarrow \left[ {13 + \left\{ {7 – 4} \right\}} \right] \times 3\) (Round brackets removed)

\(\Rightarrow[13+3] \times 3 \) (Curly brackets removed)

\(\Rightarrow 16 \times 3\) (Square brackets removed)

\(\Rightarrow 48\)

2. \(16 + \left[ {22 – \left\{ {8 + \left( {6 \div 2} \right)} \right\}} \right]\)

\(16 + \left[ {22 – \left\{ {8 + 3} \right\}} \right]\) (Round brackets removed)

\(\Rightarrow 16+[22-11]\) (Curly brackets removed)

\(\Rightarrow 16+11\) (Square brackets removed)

\(\Rightarrow 27\)

Algebraic Expressions

An algebraic expression is a combination of constants and literals/variables connected by the signs of fundamental operations like subtraction, addition, multiplication, and division.

Example: \(2 x+3\)

Constant: A constant is a number with a fixed value.

In \(2 x+3,3\) is constant.

Variable: A variable is a symbol that isn’t assigned a specific value.

In \(2 x+3, x\) is a variable.

Term: A term can consist of a single constant, a single variable, or a combination of the two.

In \(2 x+3,2 x\) and \(3\) are two terms of the expression.

Value of an Expression

The value of the expression is the result of the calculation described by this expression when the variables and constants in it are assigned values.

Letters can be used to represent numbers in an algebraic expression. The term “evaluating the expression” refers to substituting a specified value for each variable and performing the operations.

Examples:

1. Value of an expression \(3 x+5\) if \(x=2\).

\(\Rightarrow 3 x+5\)

\(\Rightarrow 3(2)+5\)

\(\Rightarrow 6+5=11\)

Therefore, the value of the given expression is \(11\).

2. Value of an expression \(8 y-4\) if \(y=3\).

\(\Rightarrow 8 y-4\)

\(\Rightarrow 8(3)-4\)

\(\Rightarrow 24-4=20\)

Therefore, the value of the given expression is \(20\).

Absolute Value

The absolute value of a number is its distance from \(0\) on the number line.
We all know that distance is usually a positive number. So the absolute value is a measure of distance. It is never negative.

The absolute value of a number \(p\) is written as \(|p|\).

\(|p| \geq 0\) for all numbers

Examples:

1. \(|6|=6\)
2. \(|-6|=6\)

Absolute Value of an Expression

Absolute values are always positive (either zero or positive). So, if you find a number in absolute value brackets, you can use the nonnegative form of that number to replace it. In general, expressions inside the modulus can be treated the same way as expressions in parentheses. Use the regular order of operations within the absolute values, apply the absolute value, and consider the outside terms.

Examples:

1. \(|12-8|=|4|=4\)
2. \(|-33+12|=|-21|=21\)

Solved Examples on Value of an Expression

Q.1. Evaluate the numerical expression \(10^{2}-20+100\).
Ans: First, evaluate the square value
\(\Rightarrow 10 \times 10-20+100\)
\(\Rightarrow 100-20+100\)
\(\Rightarrow 80+100\)
\(\Rightarrow 180\)
Hence, the obtained value of the given expression is \(180\).

Q.2. Find the value of the following expressions at \(a=1\) and \(b=-2\):
\(a^{2}+b^{2}+2 a b\).
Ans: Given \(a^{2}+b^{2}+2 a b\) where \(a=1\) and \(b=-2\).
\(\Rightarrow a^{2}+b^{2}+2 a b\)
\(=1^{2}+(-2)^{2}+2(1)(2)\)
\(=1+4+4\)
\(=9\)
Hence, the obtained value of the given expression is \(9\).

Q.3. Find the value of the expression \(\left[ {16 + \left\{ {7 – \left( {8 \div 2} \right)} \right\}} \right] \times 4.\)
Ans: Given \(\left[ {16 + \left\{ {7 – \left( {8 \div 2} \right)} \right\}} \right] \times 4.\)
\( = \left[ {16 + \left\{ {7 – 4} \right\}} \right] \times 4.\) (Round brackets removed)
\(=[16+3] \times 4 \) (Curly brackets removed)
\(=19 \times 4 \) (Square brackets removed)
\(=76\)
Hence, the obtained value of the given expression is \(76\).

Q.4. Find the value of the following expressions at \(a=2\) and \(b=-1\):
\(a^{3}+a^{2} b+a b^{2}+b^{3}\)
Ans: Given \(a^{3}+a^{2} b+a b^{2}+b^{3}\) where \(a=2\) and \(b=-1\).
\(\Rightarrow a^{3}+a^{2} b+a b^{2}+b^{3}\)
\(=2^{3}+2^{2}(-1)+(2)(-1)^{2}+(-1)^{3}\)
\(=8-4+2-1\)
\(=10-5=5\)
Hence, the obtained value of the given expression is \(5\).

Q.5. Find the values of following expressions at \(m=1, n=-1\) and \(p=2\):
1. \(m+n+p\)
2. \(m^{2}+n^{2}+p^{2}\)
Ans:
1. Given \(m+n+p\) at \(m=1, n=-1\) and \(p=2\)
\(\Rightarrow m+n+p\)
\(\Rightarrow 1+(-1)+2\)
\(\Rightarrow 0+2\)
\(\Rightarrow 2\)
Hence, the obtained value of the given expression is \(2\).
2. Given \(m^{2}+n^{2}+p^{2}\) at \(m=1, n=-1\) and \(p=2\).
\(\Rightarrow m^{2}+n^{2}+p^{2}\)
\(\Rightarrow 1^{2}+(-1)^{2}+2^{2}\)
\(\Rightarrow 1+1+4\)
\(\Rightarrow 6\)
Hence, the obtained value of the given expression is \(6\).

Summary

An expression’s value is determined by the values of the variables that make up the expression. There are various cases in which we must determine the value of an expression. Two types of expressions are arithmetic expressions and algebraic expressions. The arithmetic/numerical expression is a mathematical statement consisting of numbers and one or more operation symbols.

An algebraic expression is a combination of constants and literals/variables connected by the signs of fundamental operations. When the variables and constants in the expression are assigned values, the value of the expression is the result of the calculation given by this expression. This article includes the arithmetic and algebraic expressions, the value of an expression, the absolute value.

List of Important Algebraic Expressions & Formulas

FAQs on Value of an Expression

Q.1. How do you find the value of an expression?
Ans: When a number substitutes a variable, an algebraic expression is evaluated to find the value of the expression. To evaluate an expression, we first replace the given number for the variable in the expression, then use the order of operations to simplify the statement.

Q.2. What does the value of the expression? Give an example?
Ans: An expression’s value is calculated by the variables’ values that make up the expression.
Example: Find the value of the expression \(3 x-6\), for \(x=4\)
\(\Rightarrow 3 x-6\)
\(\Rightarrow 3(4)-6\)
\(\Rightarrow 12-6=6\)
So, The value of the given expression is \(6\).

Q.3: What is the value in an equation?
Ans: The variables’ values that make an equation true are called the equation’s value. On the other hand, solving an equation involves deciding what variables make the equation true.

Q.4. What is the value of a numerical expression?
Ans: The value obtained by solving the numerical expression is called the value of a numerical expression. The numerical expression is a mathematical statement consisting of numbers and one or more operation symbols. Addition, subtraction, multiplication, and division are examples of operation symbols.

Q.5. What is an equation vs. expression?
Ans: A number, a variable, or a combination of numbers, variables, and operation symbols make up an expression. An equal sign connects the two expressions to make an equation.

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