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