Verify the Equality: (5²+3)÷2² vs 5²+(3÷2²)

Indicate whether the equality is true or not.

^{$(5^2+3):2^2=5^2+(3:2^2)$}

In order to determine the correctness (or incorrectness) of the given equation, let's simplify both sides separately:

A. Let's start with the expression on the left side:

$(5^2+3):2^2$Let's simplify this expression while remembering the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and parentheses come before everything else, therefore we'll start by simplifying the expression inside the parentheses, this is done by calculating the numerical value of the terms with exponents within them, then we'll calculate the addition operation in the parentheses:

$(5^2+3):2^2 =\\ (25+3):2^2 =\\ 28:2^2$We'll continue and remember that exponents come before division, therefore, we'll first calculate the term with the exponent which is the divisor in the expression (in fact, if we were to convert the division operation to a fraction, this term would be in the denominator), then we'll calculate the result of the division operation:

$28:2^2 =\\ 28:4 =\\ 7$We've finished simplifying the expression on the left side of the given equation, let's summarize the simplification process:

$(5^2+3):2^2 =\\ 28:2^2= \\ 28:4 =\\ 7$B. Let's continue with the expression on the right side of the given equation:

$5^2+(3:2^2)$Similar to what we did in the previous part we'll simplify the expression while adhering to the order of operations mentioned earlier, therefore, we'll again start by simplifying the expression inside the parentheses, this is first done by calculating the numerical value of the term with the exponent (since exponents come before division), then we'll perform the division operation on the second term from the left (in parentheses), simultaneously we'll calculate the numerical value of the term with the exponent (the first from the left) and then we'll perform the addition operation:

$5^2+(3:2^2) =\\ 5^2+(3:4)=\\ 25+\frac{3}{4}=\\ 25\frac{3}{4}$Note that since the division operation yielded a non-whole number we settled for converting this operation to a fraction, finally we performed the addition operation between the whole number and the fraction and wrote the result as a mixed number, this fraction can be converted to a decimal but there's no need for that,

Note that in this expression the parentheses are actually meaningless because multiplication and division come before addition and subtraction anyway, but good practice says that if they're noted in the problem, they should be given precedence in the approach,

We've finished simplifying the expression on the right side of the equation, since the calculation is short there's no need to summarize,

Let's return then to the original equation and substitute in place of the expressions on both sides the results of the simplifications detailed in A and B in order to determine its correctness (or incorrectness):

$(5^2+3):2^2=5^2+(3:2^2) \\ \downarrow\\ 7=25\frac{3}{4}$Now we can definitively determine that the given equation is incorrect, meaning - we have a false statement,

Therefore the correct answer is answer B.