Answer:
it's all about Order of Operations (lots of caveats here)
Step-by-step explanation:
These are exercises in the use of the Order of Operations.
The Order of Operations is a set of rules for how to evaluate an arithmetic expression. It tells you which operations to perform first, so that expressions like 1+2×3 are properly evaluated to 1+6 = 7, rather than incorrectly to 3×3 = 9, for example.
(Please be aware that some calculators do this incorrectly.)
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In most cases involving numerical expressions, such as the ones shown, the rules are summarized by the mnemonics BEDMAS or BIDMAS or PEMDAS. The first two are more often used where British English is spoken. The meaning of these mnemonics is as follows:
B/P -- "brackets" or "parentheses". Expressions inside parentheses are evaluated first. Innermost parentheses are evaluated before enclosing parentheses. Your first expression shows an example of this:
4 ÷2 +5(10 -6)
has the expression (10 -6) evaluated first. The result is then the expression ...
4÷2 + 5×4
Note that in your last expression, the fraction bar effectively serves as parentheses for both the numerator and denominator. If this were written in plain text, it would be written as ...
(2^3 -5)/(15 +9)
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E/I -- "exponents" or "indices". Anything with an exponent (index) is evaluated after parentheses are eliminated. Your second expression is an example of this:
(2+3)² first has the parentheses removed so you have
5²
Then the exponent is evaluated. Of course the exponent, or index, tells you how many times 5 is a factor in the product:
5·5 = 25
As with fraction bars, the typography of a typeset expression can imply grouping (parentheses). If the expression ...
[tex]2^{x+5}[/tex]
were written in plain text, it would have to be written with parentheses around the exponent:
2^(x+5)
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D/M and M/D -- "division" and "multiplication". Here is where confusion may arise. Multiplication and division have the same priority, so are evaluated in the order they are found, left to right. The sequence of letters D or M in the mnemonic does not mean that one is performed before the other. Your first expression is an example of this:
4÷2 + 5×4 = 2 + 5×4 = 2 + 20
Here, the division happens to be first, so it is evaluated first. The multiplication is then evaluated second, because it appears to the right of the division.
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AS -- "addition" and "subtraction". Again, confusion sometimes arises because the A appears before the S in the mnemonic. The operations of addition and subtraction have the same priority, so are evaluated in the order they are found, left to right.
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These are the rules. They can be modified somewhat by the properties of multiplication and addition when you actually perform the math.
For example, the expression 2×3×5 evaluated according to the rules would be = 6×5 = 30. However, we know that multiplication has associative and commutative properties, so we could evaluate this as 2×5×3 = 10×3 = 30. Similarly, we know that multiplication and division are inverse operations, so division by 2 is the same as multiplication by (1/2). Then we could evaluate ...
4÷2 = 4×0.5 = 2
This knowledge can let us rearrange an expression such as ...
5÷2×6 = 5×(1/2)×6 = 6×(1/2)×5 = 3×5 = 15
While this is a correct evaluation of the expression, you need to be careful. If your work is supposed to show an understanding of Order of Operations, you may be required to show your work as 5÷2×6 = 2.5×6 = 15.
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Additional comment on plain-text expressions
When in doubt, add parentheses around anything you want to consider to be a unit: numerators, denominators, exponents, fractions. Here's an example where grouping can be troublesome:
[tex]\boxed{9-3\div\dfrac{1}{3}+1} = 9-3\times3+1=9-9+1=0+1=1[/tex]
(division by 1/3 is the same as multiplication by 3/1)
When written in plain text, this is often written as ...
9 - 3 ÷ 1/3 + 1
which must be evaluated as ...
9 - 3/3 +1 = 9 -1 +1 = 9 . . . . . . the division 3÷1 is performed first
The key here is that 1/3 is intended to be treated as a fraction, so it needs to be in parentheses in plain text:
9 - 3 ÷ (1/3) + 1
This last example also shows that the order of operations treats ÷ and / and "over" and "divided by" as meaning the exact same thing: whatever is on the immediate left is divided by whatever is on the immediate right.* If that "whatever" is intended to be more than a single number or variable, it must be in parentheses.
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* Yet another caveat: some authors consider "/" to always be a fraction bar, and ÷ to have the more conventional interpretation. Some authors consider ÷ to mean (everything to the left) divided by (everything to the right). The reason for the Order of Operations is to specify a standard interpretation of these symbols and their use.
