Simplification Shortcuts and Important Formulas

'BODMAS' Rule

This rule depicts the correct sequence in which the operations are to be executed, so as to find out the value of given expression.
Here,
B - Bracket,
O - of,
D - Division,
M - Multiplication,
A - Addition and
S - Subtraction
Thus, in simplifying an expression, first of all the brackets must be removed, strictly in the order (), {} and ||.
After removing the brackets, we must use the following operations strictly in the order:
(i) of (ii) Division (iii) Multiplication (iv) Addition (v) Subtraction.

Modulus of a Real Number

Modulus of a real number a is defined as

|a| = a,if a > 0

-a if a < 0

Thus, |5| = 5 and |-5| = -(-5) = 5.

Vernacular (or Bar)

When an expression contains Vernacular, before applying the 'BODMAS' rule, we simplify the expression under the Vernacular.

Roots

Roots" (or "radicals") are the "opposite" operation of applying exponents; you can "undo" a power with a radical, and a radical can "undo" a power. For instance, if you square 2, you get 4, and if you "take the square root of 4", you get 2; if you square 3, you get 9, and if you "take the square root of 9", you get 3:

$2^{2}\:=\: 4\: so\: \sqrt{4}\:=\:2$

$5^{2}\:=\:25\: so\: \sqrt{25}\: =\: 5$

b) The " $\sqrt{}$ " symbol is called the "radical"symbol. (Technically, just the "check mark" part of the symbol is the radical; the line across the top is called the "vinculum".) The expression " $\sqrt{9}$ " is read as "root nine", "radical nine", or "the square root of nine".

c) You can raise numbers to powers other than just 2; you can cube things, raise them to the fourth power, raise them to the 100th power, and so forth. In the same way, you can take the cube root of a number, the fourth root, the 100th root, and so forth. To indicate some root other than a square root, you use the same radical symbol, but you insert a number into the radical, tucking it into the "check mark" part. For instance:

$4^{3}\:=\:64 \:so\:\sqrt[3]{64}\:=\:4$

d) The "3" in the above is the "index" of the radical; the "64" is "the argument of the radical", also called "the radicand". Since most radicals you see are square roots, the index is not included on square roots.

1. a square (second) root is written as $\sqrt{}$

2. a cube (third) root is written as $3\sqrt{}$

3. a fourth root is written as $4\sqrt{}$

4. a fifth root is written as: $5\sqrt{}$

e) Then you'd round the above value to an appropriate number of decimal places and use a real-world unit or label, like "1.7 ft/sec". On the other hand, you may be solving a plain old math exercise, something with no "practical" application. Then they would almost.certainly want the "exact" value, so you'd give your answer as being simply " $\sqrt{3}$ "

Simplifying Square-Root Terms

1) To simplify a square root, you "take out" anything that is a "perfect square"; that is, you take out front anything that has two copies of the same factor: $\sqrt{4}\:=\:\sqrt{2^{2}}\:=\:2$

$\sqrt{49}\:=\:\sqrt{7^{2}}\:=\:7$

$\sqrt{225}\:=\:\sqrt{15^{2}}\:=\:15$

2) Sometimes the argument of a radical is not a perfect square, but it may "contain" a square amongst its factors. To simplify, you need to factor the argument and "take out" anything that is a square; you find anything you've got a pair of inside the radical, and you move it out front. To do this, you use the fact that you can switch between the multiplication of roots and the root of a multiplication. In other words, radicals can be manipulated similarly to powers:

$(ab)^{n}\:=\: a^{n}b^{n}\;and\;\sqrt[n]{ab}\:=\:\sqrt[n]{a}\sqrt[n]{b}$

Solved Examples for Simplification

Question 1) : The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:
Solution : Let the cost of a chair and that of a table be Rs. x and Rs. y respectively.

$y\:=\:\frac{5}{2}x$

$\therefore$ 15x + 2y = 4000

$\Rightarrow 15x+\frac{5}{2}x\times\:2\:=\:4000 \Rightarrow 20x\:=\:4000 \therefore\:x\:=\:200y\:=\:\frac{5}{4}\times 200\:=\:500$

Hence, the cost of 12 chairs and 3 tables = 12x + 3y = Rs. (2400 + 1500) = Rs. 3900.
Question 2) : If a - b = 3 and $\left( a^{2} +b^{2}\right)\:=\:29$ find the value of ab
Solution : $2ab\:=\: \left ( a^{2}+b^{2} \right )-\left ( a-b \right )^{2}$

= 29 - 9 = 20

ab=10

Question 3) : Simplify $\sqrt{4500}$
Solution : $\sqrt{4500}\: =\: \sqrt{45\times 100}\:=\: \sqrt{5\times 9\times 100}$

$3\times 10\times \sqrt{5}\:=\: 30\sqrt{5}$ (Variables in a radical's argument are simplified in the same way: whatever you've got a pair of can be taken "out front".)

Question-4) Simplify by writing with no more than one radical: $\sqrt{6}\sqrt{15}\sqrt{10}$
Solution : $\sqrt{6}\sqrt{15}\sqrt{10}\:=\:\sqrt{6\times 15\times 10}$