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Calculating a square root is about finding a number that, when multiplied by itself, yields the original number under the root symbol. In our exercise, the square root operation was applied to the number 4. Here’s how you can think of it:

- The symbol \(\sqrt{}\) represents the square root.
- Square root of 4 is asking: "What number times itself equals 4?"

When we evaluate \(\sqrt{4}\), we find that 2 satisfies this condition because \(2 \times 2 = 4\). Hence, \(\sqrt{4} = 2\).

This operation is crucial in the exercise because it helps simplify the expression \(-2^{2\sqrt{x}} + 2\). By calculating \(\sqrt{4} = 2\), we simplify the expression to become \(-2^{2 \times 2} + 2\). This simplification is an underlying skill you’ll often use in solving GMAT level math problems.

Exponents indicate how many times you multiply a number by itself. The base is the number being multiplied, and the exponent tells you how many times to do it. In the given problem, the exponent evaluation was a key step.

Here, after determining that \(\sqrt{x} = 2\), the expression becomes \(2^4\).

- Base: 2
- Exponent: 4

This means you multiply 2 by itself 4 times:

\[2 \times 2 \times 2 \times 2 = 16\]

This computation is essential to converting the expression into a numerical value which can then be used in further calculations. Understanding how to evaluate exponents quickly and accurately is a vital skill in math, especially for tests like the GMAT.

Integer operations allow you to add, subtract, multiply, or divide whole numbers. They are fundamental in solving equations and expressions. In our problem's final step, an integer operation determines the solution. The expression had boiled down to \(-16 + 2\).

This involves adding two integers: -16 and 2.
- When adding a positive integer to a negative integer, you subtract the smaller absolute value from the larger absolute value.

So, \(-16 + 2\) means taking away 2 from 16, leaving us with -14. When you're working with integers, it's crucial to be mindful of signs, as they influence the direction and outcome of the operation. Efficient integer operations are a basic but indispensable skill for solving many types of math problems.