Overview:
Parentheses in equations mean that numbers and variables should be combined first, regardless of the standard order of operations. For example, an expression such as (3 + 4)∙ 5, which equals 35, has a different meaning than 3 + (4 ∙ 5), which equals 23. Similarly, an expression such as (2 + 3)/7x has a different meaning than 2 + 3/7x.
What Is the Order of Operations?
In the standard order of operations, multiplication and division are performed before addition and subtraction. A scientific calculator will evaluate an expression such as 3 + 4 ∙ 5 by performing the multiplication first, or as 3 + 20 or 23. Similarly, 12x + 8/4 will be evaluated as 12x + 2, so that if the equation were 12x + 8/4 = 14, it would be solved as 12x = 14 – 2 , or 12x = 12, or x = 1.
Parentheses Have Priority
Parentheses mean that the expression within the parentheses must be solved first, even if those numbers wouldn’t normally be combined first. Suppose the equation were (4 + 3)x -14 = 0. If 7x = 14, then x = 2. If the parentheses were not there, the equation would be 4 + 3x – 14 = 0 or 3x – 10 = 0, combining the 4 and the -14. Then 3x = 10, so x = 3.33, or 3 1/3.
Parentheses Even Have Priority Over Exponents
Similarly, an expression such as 4x2 is very different from (4x)2. The expression 4x2means 4x ∙ x, while the expression (4x)2means 4x ∙ 4x. Suppose the equation is 4x2 = 100. Then x2 = 25, or x = 5. If the equation were (4y)2 = 160, then 16y2 = 160 or y2 = 10 , so that y = √10.
Is There an Order of Operations within Parentheses?
Suppose the expression were (13 – 2 ∙ 3)(6/3 – 1). The order of operations would be followed, so that (13 – 6) (2 – 1) or 7. If numbers were to be combined differently, there would be parentheses within parentheses or ((13 – 2)3)(6/(3-1)) or 33(3) or 99. Similarly, suppose the equation were (13 – (2x)2)(3 – 1) = 0. That would be solved by working the expressions within parentheses first, so that (13 – 4x2)(2) = 0, or 26 – 8x2 = 0, or 26 = 8x2 , or 26/8 = x2.. Since 26/8 in simplest terms is 13/4, then x = √ 13/4.
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