Question

multiply : ( 9a +14b3_15) by (12a+ 13b)

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Answer 1

Answer:

Step-by-step explanation:

I guess you don’t just want a number as the answer, but some reasons why this is evaluated as 23 and not 28.

Mathematical notation has evolved during several centuries and some conventions were developed and accepted in order to simplify the way we write expressions efficiently, but still making them readable.

One of the most ancient conventions is that multiplication has precedence over addition. This is reflected in how multiplication is denoted: in formal mathematics the “times” symbol is seldom used, preferring a centered dot or nothing at all. So  xy  and  x⋅y  mean the same thing. By the way, the reason for not using  ×  is precisely that it can be confused with  x , a letter that’s very frequently used.

When numbers are considered it would be wrong to use no symbol for denoting multiplication and, in this case, some still use  × . Thus  9⋅2  and  9×2  are equivalent, but of course using no symbol would lead to  92  which means a very different thing.

Returning to expressions, the convention is to interpret

a+bc  

as “the sum of  a  with the product  bc ” and not as “sum  a  and  b  and then multiply the result by  c ”.

Why this convention developed? It’s not easy to answer, but it came up quite naturally and is everywhere accepted. Essentially it was born when polynomials began to be used:  xxx+2xx−3x+1  is interpreted as “cube  x , then sum with twice the square of  x , then subtract three times  x  then add 1”. Later exponents were introduced, and the expression started to be written as  x3+2x2–3x+1  (and exponentiation gained stronger precedence than multiplication).

In the case we want to change the order of precedence, we use parentheses

(a+b)c  

means exactly “first sum  a  and  b  and then multiply the result by  c ”.

The accepted convention is that parenthesized expressions are evaluated first as “subexpressions”, with the same conventions holding in them.

Thus there is only one accepted interpretation of  5+9⋅2  (or  5+9×2 , if you prefer), that is

sum 5 with the product of 9 with 2

that is, first evaluate  9⋅2=18  and then do  5+18=23 .

When in doubt, use letters in place of numbers and remember this description.