Solve for the Missing Cube: (5² + 3²) ÷ (√16 · √9 + 1 + 2³/2) Expression

Let's simplify the expressions on both sides of the equation separately:

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

$(5^2+3^2):(\sqrt{16}\cdot\sqrt{9}+1+2^3:2)$Let's recall the order of operations in which powers take precedence over multiplication and division, and multiplication and division take precedence over addition and subtraction,

so let's start by simplifying the expressions inside the parentheses, where we will first calculate the values of the terms in the power and in the roots (which are powers of everything), then we will calculate the values of the products and results of the division operations, and finally we will perform the addition operations in the parentheses:

$(5^2+3^2):(\sqrt{16}\cdot\sqrt{9}+1+2^3:2) =\\ (25+9):(4\cdot3+1+8:2) =\\ 34:(12+1+4)=\\ 34:17=\\ 2$In the last step we calculated the result of the division operation that remained (which was originally between the two main parentheses),

We have thus completed the simplification of the expression on the left side.

B. Let's continue simplifying the expression on the right side:

$\frac{(6^2-\sqrt{16}):\textcolor{red}{☐}^3}{\sqrt{4}}$For convenience of operations, let's call this unknown number we are looking for, let's define it as- x:

$\textcolor{red}{☐}=x\\ \downarrow\\ \textcolor{red}{☐}^3=x^3\\$and we will place it in the mentioned expression:

$\frac{(6^2-\sqrt{16}):\textcolor{red}{☐}^3}{\sqrt{4}} \\ \downarrow\\ \frac{(6^2-\sqrt{16}):x^3}{\sqrt{4}}$Let's continue simplifying the expression, let's start by calculating their numerical value of the terms in the power and in the root in the numerator, in parallel we will calculate the numerical value of the term in the root in the denominator, then we will calculate the result of the subtraction operation in the numerator:

$\frac{(6^2-\sqrt{16}):x^3}{\sqrt{4}}= \\ \frac{(36-4):x^3}{2} =\\ \frac{32:x^3}{2}$For convenience of solution for now, we will leave this expression in its current form and emphasize that we have completed the treatment of the expression on the right side.

Let's go back to the original equation and place in it the two simplification results we got for the terms on the left and right sides that were detailed in A and B, let's not forget that we also defined the number we are looking for as x:

$(5^2+3^2):(\sqrt{16}\cdot\sqrt{9}+1+2^3:2)=\frac{(6^2-\sqrt{16}):\textcolor{red}{☐}^3}{\sqrt{4}} \\ \downarrow\\ (5^2+3^2):(\sqrt{16}\cdot\sqrt{9}+1+2^3:2)=\frac{(6^2-\sqrt{16}):x^3}{\sqrt{4}} \\ \downarrow\\ 2= \frac{32:x^3}{2} \\ \frac{2}{1}= \frac{32:x^3}{2}$$$In the last step we used the fact that any number can be written as a number divided by 1, we did this as a preparation for the next step where we will solve the equation that was obtained by multiplying both sides by the common denominator,

Now let's simplify the equation we got, again we will multiply both sides by the common denominator, but this time the common denominator is the algebraic expression: $x^3$, this expression depends on the unknown we are looking for and therefore we must define a permitted range of definition, since multiplying the equation by 0 is forbidden (and in general division by 0 is also a forbidden operation, so from the beginning we can define this range of definition):

$\frac{4}{1}=\frac{32}{x^3} \hspace{8pt}\text{/}\cdot x^3\hspace{4pt};x\neq0\\ 4 x^3=32$Again- we will know how much to multiply each term by using the answer to the question:"By how much did we multiply the current denominator in order to get the common denominator?", we also did not forget to define the permitted range of definition (we got it by solving the inequality:

$x^3\neq0$)

Let's continue and isolate the unknown by dividing both sides of the equation by its coefficient:

$4 x^3=32 \hspace{8pt}\text{/}:4\\ x^3=\frac{\not{32}}{\not{4}}\\ x^3=8$When in the last step we reduced the fraction that was obtained on the right side as a result of the division operation,

Let's finish solving the equation by taking the cube root of both sides of the equation, the cube root which is an operation inverse to the cube power will cancel the cube power on the unknown on the left side of the equation:

$x^3=8 \hspace{8pt}\text{/}\sqrt[3]{\hspace{4pt}}\\ \sqrt[3]{x^3}=\sqrt[3]{8}\\ x=2$Note that since we took an odd-order root from both sides of the equation we only need to consider one solution (it is the solution we will get in the calculator when connecting the cube root of the number 8) as opposed to taking an even-order root, in which case we need to consider two possible solutions - a positive solution and a negative solution,

We have thus solved the equation, and we got that the number we are looking for, which we defined to be the unknown x (marked at the beginning of the problem in a red square) is the number 2,

So the correct answer is answer D.