Answer:
Explanation:
You want to evaluate two numerical expressions.
The order of operations applies. In order of precedence, we evaluate parentheses, exponents, {multiplication and division}, {addition and subtraction}.
Exponents are evaluated right-to-left. Multiplication and division are evaluated left-to-right. Addition and subtraction are evaluated left-to-right. (Both multiplication and addition can be rearranged according to their respective commutative and associative properties.)
In the working below, each bold number is the value of the expression it replaces.
a)
(54-(15+14*21/7-75)^2)/(47*5-510/6+(-1560/13))
We identify any multiplication and division inside the innermost parentheses, and do that first.
= (54-(15+294/7-75)^2)/(47*5-510/6+(-1560/13))
= (54-(15+42-75)^2)/(47*5-510/6+(-1560/13))
= (54-(57-75)^2)/(47*5-510/6+(-1560/13))
= (54-(-18)^2)/(47*5-510/6+(-1560/13))
= (54-324)/(47*5-510/6+(-1560/13)) . . . . . evaluate exponent
= (54-324)/(47*5-510/6-120) . . . . . other innermost parentheses
= (54-324)/(235-510/6-120) . . . . . division
= (54-324)/(235-85-120)
= -270/(235-85-120) . . . . . subtraction (outer parentheses)
= -270/(150-120) . . . . . . . subtraction (other outer parentheses)
= -270/30
= -9
b)
14-(7+4*3-((-2)^2*2-6))+(2^2+6-5*3)+3-(5-2^3/2)
The innermost parentheses involve an exponent, so we do that first. Then we can evaluate parentheses that are at the same level left to right.
= 14-(7+4*3-((-2)^2*2-6))+(2^2+6-5*3)+3-(5-2^3/2)
= 14-(7+4*3-(4*2-6))+(2^2+6-5*3)+3-(5-2^3/2) . . . . exponent
= 14-(7+4*3-(8-6))+(2^2+6-5*3)+3-(5-2^3/2) . . . . . inner parentheses
= 14-(7+4*3-2)+(2^2+6-5*3)+3-(5-2^3/2)
= 14-(7+12-2)+(2^2+6-5*3)+3-(5-2^3/2) . . . . . multiplication
= 14-(19-2)+(2^2+6-5*3)+3-(5-2^3/2)
= 14-17+(2^2+6-5*3)+3-(5-2^3/2)
= 14-17+(4+6-5*3)+3-(5-2^3/2) . . . . . next parentheses, exponentiation
= 14-17+(4+6-15)+3-(5-2^3/2)
= 14-17+(10-15)+3-(5-2^3/2)
= 14-17-5+3-(5-2^3/2)
= 14-17-5+3-(5-8/2) . . . . . next parentheses, exponentiation
= 14-17-5+3-(5-4)
= 14-17-5+3-1
= -3-5+3-1
= -8+3-1
= -5-1
= -6
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Additional comment
A suitable calculator can evaluate these expressions according to the order of operations.
About exponents
The rules of exponents tell you ...
(a^b)^c = a^(bc)
The order of operations tells you ...
a^b^c = a^(b^c)
You will notice these are different, which is why parentheses are required in the first case.
Parentheses are required whenever there is any arithmetic in an exponent. For example, ...
a^1/2 = (a^1)/2 = a/2
a^(1/2) = √a