Complex Fraction Equation: Find the Missing Square Number in (-1)^8 Expression

Let's simplify the equation, dealing with the fractions in both sides separately:

A. We'll start with the fraction on the left side:

$\frac{(-1)^8-(-3)\cdot\sqrt{16}+3}{6^2:(-3)^2-2^2\cdot2}$Let's simplify this fraction while remembering the order of operations, meaning that exponents come before division and multiplication, which come before addition and subtraction, and parentheses come before everything,

Additionally, we'll remember that an exponent is multiplying a number by itself and therefore raising any number (even a negative number) to an even power will give a positive result, and this is because negative one multiplied by negative one gives the result of one,

In particular, in this problem:

$(-1)^8=1\\ (-3)^2=9$Let's return to the fraction and apply this, first we'll deal with the numerator of the fraction where we'll calculate the numerical value of the square root in the second term from the left and the value of the term with the exponent:

$\frac{(-1)^8-(-3)\cdot\sqrt{16}+3}{6^2:(-3)^2-2^2\cdot2}=\\ \frac{1-(-3)\cdot4+3}{6^2:(-3)^2-2^2\cdot2}$Now we'll remember that according to the multiplication rules, multiplying a negative number by a negative number will give a positive result, we'll apply this to the numerator of the fraction in question, then we'll perform the multiplication in the second term from the left and simplify the expression in the numerator of the fraction:

$\frac{1-(-3)\cdot4+3}{6^2:(-3)^2-2^2\cdot2} =\\ \frac{1+3\cdot4+3}{6^2:(-3)^2-2^2\cdot2} =\\ \frac{1+12+3}{6^2:(-3)^2-2^2\cdot2}=\\ \frac{16}{6^2:(-3)^2-2^2\cdot2}$We'll continue and simplify the denominator of the fraction, we'll start by calculating the values of the terms with exponents, this we'll do, again, using what was said before (regarding the even exponent), then we'll perform the division operation in the first term from the left and the multiplication operation in the second term from the left, and finally we'll perform the subtraction operation in the denominator of the fraction:

$\frac{16}{6^2:(-3)^2-2^2\cdot2} =\\ \frac{16}{36:9-4\cdot2}=\\ \frac{16}{4-8}=\\ \frac{16}{-4}=\\ -4$In the last stage we performed the division operation of the fraction, this while we remember that according to the division rules (which are identical to the multiplication rules in this context) dividing a positive number by a negative number will give a negative result,

We have finished simplifying the fraction on the left side, let's summarize this simplification:

$\frac{(-1)^8-(-3)\cdot\sqrt{16}+3}{6^2:(-3)^2-2^2\cdot2}=\\ \frac{1-(-3)\cdot4+3}{6^2:(-3)^2-2^2\cdot2} =\\\ \frac{16}{6^2:(-3)^2-2^2\cdot2} =\\ \frac{16}{36:9-4\cdot2}=\\ \frac{16}{-4}=\\ -4$

B. Let's simplify the fraction on the right side:

$\frac{(-2)^3+\textcolor{red}{☐}^2-1}{-(-2)^2}$For convenience, let's name the number we're looking for and define it as x:

$\textcolor{red}{☐}=x\\ \downarrow\\ \textcolor{red}{☐}^2=x^2$Let's substitute this in the fraction in question:

$\frac{(-2)^3+x^2-1}{-(-2)^2}$So let's start simplifying the fraction, again we'll use the fact that raising any number to an even power will give a positive result, but we'll also remember the fact that raising a negative number to an odd power will give a negative result, in particular in the problem:

$(-2)^2=4\\ (-2)^3=-8$Let's apply this to the fraction in question:

$\frac{(-2)^3+x^2-1}{-(-2)^2} =\\ \frac{-8+x^2-1}{-4}$Now let's combine like terms in the numerator of the fraction:

$\frac{-8+x^2-1}{-4}=\\ \frac{-9+x^2}{-4}$We have finished simplifying the fraction on the right side of the given equation, let's summarize this simplification:

$\frac{(-2)^3+x^2-1}{-(-2)^2} =\\ \frac{-8+x^2-1}{-4} =\\ \frac{-9+x^2}{-4}$

Let's now return to the original equation and substitute the results of simplifying the fractions on the left and right sides detailed in A and B respectively, we won't forget the definition of the unknown x mentioned earlier:

$\frac{(-1)^8-(-3)\cdot\sqrt{16}+3}{6^2:(-3)^2-2^2\cdot2}=\frac{(-2)^3+\textcolor{red}{☐}^2-1}{-(-2)^2} \\ \downarrow\\ \frac{(-1)^8-(-3)\cdot\sqrt{16}+3}{6^2:(-3)^2-2^2\cdot2}=\frac{(-2)^3+x^2-1}{-(-2)^2} \\ \downarrow\\ -4= \frac{-9+x^2}{-4}$ Let's continue and solve the resulting equation, first we'll remember that any number can be represented as itself divided by the number 1:

$-4= \frac{-9+x^2}{-4}\\ \frac{-4}{1}= \frac{-9+x^2}{-4}$Then we'll multiply both sides of the equation by the common denominator, which is the number $-4$ and then we'll simplify the equation and isolate the unknown by moving terms and combining like terms:

$\frac{-4}{1}= \frac{-9+x^2}{-4} \hspace{8pt}\text{/}\cdot(-4) \\ (-4)\cdot(-4)=-9+x^2\\ 16=-9+x^2\\ 16+9=x^2\\ 25=x^2$In the first stage we multiplied by the common denominator in order to cancel the fraction line while we multiply each numerator by the number that is the answer to the question "By how much did we multiply the current denominator to get the common denominator?" ,

Let's return to what was asked, we are looking for the number x (in the original problem it is a red square) for which we get a true statement from the last equation, we can take the square root of both sides of the equation and get the two possible solutions (we'll remember that taking a square root in the context of solving an equation always involves taking into account two possibilities, positive and negative) but we can also use logic and remember that raising any number to an even power will give a positive result, meaning the above equation has two possible answers, a positive answer and a negative answer,

We'll add and ask: "What number did we raise to the second power to get the number 25?" a question to which the answer is of course the number 5 (or minus 5), and therefore the full answer is:

$x=5,\hspace{4pt}-5$ Each of these options is correct,

And therefore the correct answer is answer C.