Introduction
Simplification in mathematics involves reducing an expression or equation to its simplest form. It could involve solving operations in brackets, simplifying roots and exponents, or reducing fractions to their lowest terms. By simplifying, we can make complex mathematical problems more manageable.
Importance in SSC, RRB, and Banking Exams
Simplification forms the basis of many questions that appear in SSC, RRB, and banking exams. Often, numerical ability or quantitative aptitude sections include problems that require simplification. A solid understanding of simplification rules can be instrumental in solving these problems efficiently and accurately.
Basic Concepts and Definitions
Here are some basic concepts that are important in simplification:
Order of Operations: Also known as BODMAS or PEMDAS, it stands for Brackets, Orders (powers and roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). It dictates the order in which operations should be performed to accurately simplify expressions.
Fractions: These can often be simplified by reducing to their lowest terms, that is, dividing the numerator and denominator by their greatest common divisor.
Square Roots and Cube Roots: These can often be simplified by identifying factors that are perfect squares or cubes.
Exponents: These can be simplified using the laws of exponents.
Simplification Rules and Techniques
Using the BODMAS/PEMDAS rule, one can solve complex mathematical expressions. It’s essential to keep the order to avoid any miscalculations. For fractions, always try to reduce them to their lowest terms. When dealing with roots and exponents, it’s useful to memorize the squares, cubes, square roots, and cube roots of numbers up to at least 15.
Problem-Solving Techniques and Examples
Here are a few examples of simplification problems:
Example 1: Simplify the expression (7 + 3) * 2 – 4^2.
According to the BODMAS rule, we first solve the brackets (7+3) to get 10, then the power 4^2 to get 16. So, our equation now reads 10 * 2 – 16. Next, we perform the multiplication to get 20 – 16, which finally simplifies to 4.
Example 2: Simplify the fraction 18/48.
The greatest common divisor of 18 and 48 is 6. So, when we divide both the numerator and the denominator by 6, the fraction simplifies to 3/8.
Practice Problems
Try simplifying these:
- (3 + 5) * 2^3 – 5^2
- The fraction 81/135
- √169
Summary
Simplification is a fundamental skill in mathematics. It not only helps to solve complex mathematical problems easily but also plays a significant role in numerical ability or quantitative aptitude sections of various competitive exams. Understanding the order of operations and other simplification rules are crucial for mastering this topic. Regular practice will help you become more proficient in simplification and improve your problem-solving speed.