Determine the Sign: (10²:√16-2²·6)¹⁰⁰ vs (7:√49)+3²-2³

To solve the problem, we need to carefully evaluate both expressions on either side of the _ (underscored) red dash, $\textcolor{red}{—}$. Let's go through the calculations step-by-step, using the order of operations.

Step 1: Evaluate each component in the expressions separately:

• $10^2$: 10 squared is 100.

• $\sqrt{16}$: The square root of 16 is 4.

• $2^2$: 2 squared is 4.

• $\sqrt{49}$: The square root of 49 is 7.

• $3^2$: 3 squared is 9.

• $2^3$: 2 cubed is 8.

Step 2: Substitute the evaluated components back into the expression and reduce:

• Left Expression: $(100 : 4 - 4 \cdot 6)^{100}$

  Calculate inside the parentheses using division first: $100 : 4 = 25$

  Then calculate the multiplication: $4 \cdot 6 = 24$

  Subtract the results: $25 - 24 = 1$

  Raise to the 100th power: $1^{100} = 1$

• Right Expression: $7 : 7 + 9 - 8$

  Calculate the division: $7 : 7 = 1$

  Then add: $1 + 9 = 10$

  Finally, subtract: $10 - 8 = 2$

After evaluating, we're left with comparing $1$ on the left side and $2$ on the right side.

Conclusion: $1 \lt 2$. Therefore, the appropriate sign to mark is $\lt$.