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A perfect square is a number that is the square of an integer. For example, when you square the number 2, you get 4, which is a perfect square. Recognizing perfect squares is key in simplifying square roots, as they help you reduce the complexity of an expression. For instance:

• The number 4 is a perfect square because it's equal to \(2^2\).
• The number 100 is a perfect square since it equals \(10^2\).
• Similarly, 49 is a perfect square because it's \(7^2\).

When simplifying expressions with square roots, identifying perfect squares allows you to "take" the square root easily and include only the integer in the expression. In the original exercise, you take the square roots of 4, 100, and 49 first, simplifying the expression before proceeding to the next step.

Combining like terms is a fundamental concept in algebra, essential for simplifying expressions. This process involves summing or subtracting coefficients of terms that have the same variable part. For square root expressions like the one in the exercise, terms with the same radical component can be combined.
Consider the expression \(2\sqrt{x} - 10\sqrt{x} - 7\sqrt{x}\). Here, each term contains \(\sqrt{x}\), the square root term in common. Therefore, you can add or subtract the coefficients:

• Start with \(2\sqrt{x}\).
• Subtract \(10\sqrt{x}\), which alters the coefficient to \(-8\).
• Subtract \(7\sqrt{x}\), leading to \(-15\sqrt{x}\).

This simplification yields the final term \(-15\sqrt{x}\). Understanding this concept aids in streamlining calculations and solving expressions more efficiently.

Radical expressions involve roots, such as square roots, cube roots, etc., and they are a crucial part of many algebraic operations. A square root expression includes the radical sign \(\sqrt{}\) followed by a number or variable, which indicates the operation of finding which number, when squared, produces the number or the variable under the radical. For example, \(\sqrt{16}\) means "what number squared equals 16?" The answer would be 4.
In our exercise, we encounter terms like \(\sqrt{4x}\), \(\sqrt{100x}\), and \(\sqrt{49x}\). These incorporate both perfect squares and variable components, complicating the potential simplification. By understanding both perfect squares and radicals, you can break down complicated expressions into more manageable terms. Always remember to simplify the number part as much as practical first, which often makes the variable part simpler to handle later.

• Simplify expressions containing perfect squares first.
• Combine like terms sharing the same radical smoothly.
• Maintain the radical part for terms that aren't perfect squares.

This strategy not only assists in achieving simplicity but also clarifies the expression's original structure.