Verify the Equality: 3^4÷(√25+2^2)-2^3÷√4 vs Alternative Expression

To determine if the given equation is correct, we will simplify each expression in its sides separately,

This will be done while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

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

$3^4:(\sqrt{25}+2^2)-2^3:\sqrt{4}$We'll begin by simplifying the expressions inside the parentheses, this is done by calculating the numerical value of the terms with exponents (while remembering the definition of a root as an exponent stating that a root is actually an exponent), simultaneously we'll calculate the numerical value of the terms with exponents in the second term from the left and in the first term from the left, which is the divisor in this term (in fact, if we were to convert this expression to a simple fraction, this term would be in the fraction's numerator):

$3^4:(\sqrt{25}+2^2)-2^3:\sqrt{4} =\\ 81:(5+4)-8:2$We'll continue and finish simplifying the expression inside the parentheses, meaning we'll perform the addition operation within them, then we'll remember again the order of operations, meaning that division comes before subtraction and therefore we'll perform the division operations in both terms of the expression, in the final stage we'll perform the subtraction operation:

$81:(5+4)-8:2 =\\ 81:9-8:2 =\\ 9-4=\\ 5$We have finished simplifying the expression on the left side of the given equation, let's summarize the simplification steps:

$3^4:(\sqrt{25}+2^2)-2^3:\sqrt{4} =\\ 81:(5+4)-8:2 =\\ 9-4=\\ 5$

B. Let's continue with simplifying the expression on the right side of the given equation:

$3^4:\sqrt{25}+(2^2-2^3):\sqrt{4}$We'll start again by simplifying the expressions inside the parentheses, this is done by calculating the numerical value of the terms with exponents in the parentheses, simultaneously we'll calculate the numerical value of the other terms with exponents in the given expression (while remembering the definition of a root as an exponent stating that a root is actually an exponent):

$3^4:\sqrt{25}+(2^2-2^3):\sqrt{4} =\\ 81:5+(4-8):2$We'll continue and finish simplifying the expression inside the parentheses, meaning we'll perform the subtraction operation within them, then we'll remember again the order of operations, meaning that division comes before addition and therefore we'll perform the division operations in both terms of the expression, in the final stage we'll perform the addition operation:

$81:5+(4-8):2 =\\ 81:5+(-4):2 =\\ 81:5-4:2=\\ 20\frac{1}{5}-2\\ 18\frac{1}{5}$Note that the result of the subtraction operation in the parentheses yielded a negative result and therefore in the next step we kept this result in parentheses and then applied the multiplication law stating that multiplying a positive number by a negative number gives a negative result (so ultimately we get a subtraction operation), then, since the division operation performed in the first term from the left yielded a non-whole result (actually greater than a whole number) we wrote this result as a mixed fraction,

We have finished simplifying the expression on the right side of the given equation, let's summarize the simplification steps:

$3^4:\sqrt{25}+(2^2-2^3):\sqrt{4} =\\ 81:5+(4-8):2 =\\ 81:5-4:2=\\ 18\frac{1}{5}$Let's now return to the given equation and substitute in its sides the results of simplifying the expressions detailed in A and B:

$3^4:(\sqrt{25}+2^2)-2^3:\sqrt{4}=3^4:\sqrt{25}+(2^2-2^3):\sqrt{4} \\ \downarrow\\ 5=18\frac{1}{5}$Obviously this equation does not hold, meaning - we got a false statement,

Therefore the correct answer is answer B.