Brackets/Parenthesis and their applications:
Brackets are the symbols used for grouping the terms in an algebraic expression and they help to identify the order / sequence in which the operations are to be carried out while simplifying the expression as well as solving the equations. They are widely used when we have complex expressions with multiple terms, requiring a number of operations to be carried out to arrive at the end result. They help us to carry out the operations in an organized manner.
Parentheses, brackets, braces and radicals are some of the algebraic grouping symbols and they help us to identify where a group starts and ends and they indicate / guide the order in which the operations have to be carried out for simplification or solution. While doing so, we have to ensure that the terms inside a grouping symbol must be simplified first before allowing them to be acted upon by terms outside the grouping symbol. We must also note that all the bracket types have equal weight and none is more powerful or acts differently from others. The following group symbols are mainly used in Algebra in framing expressions and equations: –
- Parentheses (): Parentheses are the most commonly used symbols for grouping. They are also known as simple brackets.
- Brackets [] and Braces (}: When the no., of groupings is more, we resort to using brackets and braces for grouping and they have the same effect as parentheses.
- Radical √: This symbol is used to represent the square root of a number / variable and is called as the Square Root symbol.
- Fraction Line (Vinculum) —: The fraction line is another grouping symbol and helps to represent the numerator and denominator in a fraction. All the terms in the numerator are grouped and simplified to arrive at the minimum terms in the numerator – so also all the terms in the denominator are grouped and simplified to arrive at the minimum terms in the denominator to arrive at the final fraction.
Example 1. Simplify: – 3 (p +q) + 2 (q – P).
This is a simple problem involving parentheses – to simplify, remove the parentheses and proceed.
We get 3 (p +q) + 2 (q – P) = 3. p + 3. q + 2. q – 2. p
= 3 p – 2 p + 3.q – 2. q
= p + q or (p + q) is the answer.
Example 2. Simplify: 4 (2 p – q) + 3 (2 q – 5) – 2 (p + q) – 6 (q – 1)
To simplify, remove the parentheses and expand the terms and group the like terms.
We get, 4 (2 p – q) + 3 (2 q – 5) – 2 (p + q) – 6 (q – 1)
= 8 p – 4q + 6 q – 15 – 2 p – 2 q – 6 q + 6
= 8 p – 2 p – 4 q + 6 q – 2 q – 6 q – 15 + 6
= 6 p – 6 q – 9
= 3 (2 p – 2 q – 3).
Note: we must present the answer after regrouping all the like terms and bring out the common factor in all the terms.
Example 3.
Simplify the expression [3 (x + 2 y) – (2 (x – 3) – 4 (2 – y)) – 9] and find the value of (x + 2 y) – (2 (x – 3) – 4 (2 – y)) – 9] = 0 when (x + 2 y) = – 5.
The given expression
[3 (x + 2 y) – (2 (x – 3) – 4 (2 – y)) – 9] = [3 x + 6 y – (2 x – 6 – 8 + 4 y) – 9] = [3 x + 6 y – 2 x + 6 + 8 – 4 y – 9]
= [x + 2 y + 1 4 – 9]
= [(x + 2 y) + 5].
Given (x + 2 y) = – 5, the value of the above expression – 5 + 5 = 0.
Simplify the following:
1.(4 x – 3 y) + (3 z – 2 x) – (2 y + 3 z);
2. (5 a + 4 b – 2 c) – (3 a + 2 c – b);
3. (p + q – r) – (q – r + 2 p) + (2 r – p – 2 q);
4. (5 x – 3 y) + (3 z – 2 x) – (2 y + 3 z);
5. 2 (2 p + q) – 5 (p – q + 3 r) – (2 p + q)
6. 5 a – ((2 b + 3 c) – 3(a + b – 2 c)) + 3 b + 5 c
7. 5 x – 3 y) – ((2 y – 3 z + 4 x) – 2 (2 x + 3 y – z)) – 4 x; Also, find its value when (x + y + z) = 0;
8. (2 m + 3 n – 2 p) + (4 n – 3 p – m) – (5 p + 3 m – 2 n) + 3 (2 m – n + 2 p) – (m – p); If (m + 2 n) = p find the value of the expression.
9. (a + b) – ((3 c + 2 a) – 2 (b + 2 c – a)) – ((a + b – 2 c) – (2 b – 3 c + a)) + (5 a – 2 b). Simplify
10. (4 (p + 2 q + 3 r) – 3 (4 p – 3 q – 2 r)) + 2 (3 p – 2 q – 4 r) + (4 p + 3 q + 2 r)
Answers:
1. (2 x – 5 y)
2. (2 a + 5 b – 4 c)
3. – 2 (p + q – r)
4. (3 x – 5 y)
5. -3p +6q – 15r
6. 4 (2 a + b – c)
7. (x +y + z), 0
8. 0
9. 2 (a + b)
10). 2 p + 16q + 12 r.