Find the Missing Square Number in (2³+5²)-9²:3²-√100=(6·5-□²)²-√25-6

To solve the problem, we need to evaluate both sides of the equation step by step and determine which number should replace the missing value to maintain equality. We'll start by evaluating the left side and simplifying the right side to find the missing number.

Let's examine the left side of the equation:
$(2^3+5^2)-9^2:3^2-\sqrt{100}$

• First, calculate $2^3$ which is $8$.

• Next, calculate $5^2$ which is $25$.

• Add these results together: $8 + 25 = 33$.

• Calculate $9^2$ which is $81$.

• Calculate $3^2$ which is $9$.

• Divide $81$ by $9$:$\ 81 \div 9 = 9$.

• Subtract $9$ from $33$: $33 - 9 = 24$.

• Calculate $\sqrt{100}$ which is $10$.

• Subtract $10$ from $24$:$24 - 10 = 14$.

Now, let's simplify the right side of the equation:
$(6\cdot5-\textcolor{red}{☐}^2)^2-\sqrt{25}-6$

• Calculate $6\cdot5$ which is $30$.

• The expression becomes: $(30-\textcolor{red}{☐}^2)^2-\sqrt{25}-6$.

• We know from the original problem statement that the complete expression must equal $14$.

• Calculate $\sqrt{25}$, which is $5$.

• The equation: $(30-\textcolor{red}{☐}^2)^2-5-6=14$.

• Simplify further:
  $((30-\textcolor{red}{☐}^2)^2=25$

• Taking square roots of both sides: $30-\textcolor{red}{☐}^2=5$ or $-(30-\textcolor{red}{☐}^2)=5$

• Solving for $\textcolor{red}{☐}$:

  • $30-\textcolor{red}{☐}^2=5\rightarrow \textcolor{red}{☐}^2=25\rightarrow \textcolor{red}{☐}=5$

  • Alternatively, for the negative case: $-(30-\textcolor{red}{☐}^2)=5\rightarrow 30-\textcolor{red}{☐}^2=-5$ does not result in a real number solution.

Therefore, $\textcolor{red}{☐}$ must be $5$ to balance the equation