An algebraic expression is a mathematical expression formed by the combination of variables, constants, and mathematical operations
Algebraic Operations are mathematical processes performed on algebraic expressions containing variables, constants, and coefficients.
The four basic algebraic operations are the following:
1. Addition
Addition of algebraic expressions involves combining like terms to form a simplified expression.
Example: (3x+4) + (2x+5)
Combine like terms: 3x + 2x + 4 + 5 = 5x + 9
2. Subtraction
Subtraction involves removing one algebraic expression from another by changing the signs of the terms being subtracted.
Example: (7x+8) − (3x+2)
Remove brackets: 7x + 8 − 3x − 2
Combine like terms: 4x + 6
3. Multiplication
Multiplication involves multiplying coefficients and variables according to the laws of exponents.
Example: (3x)(4x2)
Multiply coefficients: 3×4 = 12
Multiply variables: x * x2 = x3
Therefore, (3x)(4x2) = 12x3
4. Division
Division involves dividing coefficients and subtracting exponents of like variables.
Example:
\frac{18x^5}{3x^2} Divide coefficients: 18÷3 = 6
Subtract exponents: x5−2 = x3
Therefore, 1
\frac{18x^5}{3x^2}=6x^3
Example: Solve 8 + 4 × (6 − 2)²
Step 1: Solve brackets (6 − 2) = 4
Step 2: Solve powers 4² = 16
Step 3: Multiply 4 × 16 = 64
Step 4: Add 8 + 64 = 72
Therefore, 8 + 4 × (6 − 2)² = 72
Solved Examples
Example 1: Simplify: (4x + 7) + (3x + 5)
Step 1: Remove the brackets.: 4x + 7 + 3x + 5
Step 2: Combine like terms: (4x + 3x) + (7 + 5) = 7x + 12
Example 2: Simplify: (8x + 10) - (3x + 4)
Step 1: Change the signs of the terms inside the second bracket: 8x + 10 - 3x - 4
Step 2: Combine like terms: (8x - 3x) + (10 - 4) = 5x + 6
Example 3: Simplify: (2x)(5x2)
Step 1: Multiply the coefficients: 2 × 5 = 10
Step 2: Multiply the variables: x × x2 = x3
Step 3: Write the final expression. 10x3
Example 4: Simplify:
Step 1: Divide the coefficients: 24 ÷ 6 = 4
Step 2: Apply the exponent rule:
x^4 \div x=x^{4-1}=x^3 Step 3: Write the simplified expression. 4x3
Example 5: Simplify: 3(2x+4)-5+x
Step 1: Solve the expression inside the bracket using the distributive property: 3(2x+4)=6x+12
Step 2: Substitute the result into the original expression: 6x+12-5+x
Step 3: Combine like terms: (6x+x)+(12-5) = 7x+7
Practice Problems
Problem 1: Simplify: (6x + 9) + (4x + 7)
Problem 2: Simplify: (12y + 8) - (5y + 3)
Problem 3: Simplify: (3a2)(4a3)
Problem 4: Expand and simplify: 5(x + 6)
Problem 5: Simplify: 2(3x - 2) + 4x - 5