Simplification: Introduction


Important Algebric Formulae: $$ (a + b)^2 = a^2 + 2ab + b^2 $$ $$ (a - b)^2 = a^2 - 2ab + b^2 $$ $$ (a^2 + b^2) = (a - b)^2 + 2ab $$ $$ (a^2 - b^2) = (a + b) \ (a - b) $$ $$ (a + b)^3 = a^3 + b^3 + 3ab \ (a + b) $$ $$ (a - b)^3 = a^3 - b^3 - 3ab \ (a - b) $$ $$ (a^3 + b^3) = (a + b) \ (a^2 + b^2 - ab) $$ $$ (a^3 - b^3) = (a - b) \ (a^2 + b^2 + ab) $$

Simplification Sequence: For simplifying a question you must follow the sequence. Remember the word "VBODMAS". $$ 1 \longrightarrow V \longrightarrow {Vinculum \ or \ Bar \ (\bar{N})} $$ $$ 2 \longrightarrow B \longrightarrow Brackets, \ (1), \left\{2\right\}, [3] $$ $$ O \longrightarrow {of} $$ $$ 3 \longrightarrow D \longrightarrow {Division (\div)} $$ $$ 4 \longrightarrow M \longrightarrow {Multiplication (\times)} $$ $$ 5 \longrightarrow A \longrightarrow {Addition (+)} $$ $$ 6 \longrightarrow S \longrightarrow {Subtraction (-)} $$

Under the Vinculum or bar \((\bar{N})\), Here N is any number

Under the brackets, solve brackets as numbered inside the brackets \((1), \left\{2\right\}, [3]\).

Addition and Subtraction: There are no shortcuts to do addition and subtraction of numbers. You must practice questions as much you can without using a calculator. The addition and subtraction for mixed fractions will be discussed in the next chapter.

Example(1): \(\left[(6)^3 \times (3)^2\right] \div (2)^3 = ?\)

Solution: $$ \left[(6)^3 \times (3)^2\right] \div (2)^3 $$ $$ \left[216 \times 9\right] \div 8 $$ $$ 1944 \div 8 = 243 $$

Example(2): \(216 \div 36 + 4 - 2 = ?\)

Solution: $$ 216 \div 36 + 4 - 2 $$ $$ 6 + 4 - 2 $$ $$ 10 - 2 = 8 $$

Example(3): \(\left[{(20)^3 \div (8 + 2)^2 \times 2}\right] + 12 - 8 = ?\)

Solution: $$ \left[{(20)^3 \div (8 + 2)^2 \times 2}\right] + 12 - 8 $$ $$ \left[{8000 \div 100 \times 2}\right] + 12 - 8 $$ $$ \left[{80 \times 2}\right] + 12 - 8 $$ $$ \left[160\right] + 12 - 8 $$ $$ 160 + 12 - 8 $$ $$ 172 - 8 = 164 $$




Simplification

Chapter 1: Introduction

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