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A "perfect square" is a number that results from an integer multiplied by itself. To determine if a number is a perfect square, we look for this exact product of repeating factors. For example, in our case with the number 4900, we discover its perfect square status by identifying the repeating factor 70 from the series of factor pairs.

Recognizing a perfect square can be extremely beneficial when simplifying square roots or dealing with algebraic expressions involving exponents. It saves time as we can directly find the root without lengthy calculations. In mathematics, perfect squares have a special place as they often signal geometric or symmetrical properties in problems.

"Factorization" involves breaking down a number into its prime factors or equal parts. It’s like reverse assembling a puzzle. When dealing with perfect squares like 4900, we factorize to find pairs of numbers that multiply to give us the original number.

In the case of 4900, by listing all factor pairs, such as 1 and 4900, 7 and 700, and eventually 70 and 70, we pinpoint 70 as the repeating factor. This consistent factor denotes that 4900 is indeed a perfect square. Factorization is a powerful tool in mathematics for simplifying expressions and solving equations efficiently. Additionally, understanding factorization strengthens skills in division and multiplication, both critical arithmetic operations.

Square roots can often be surprising, as they offer both positive and negative solutions. This duality stems from the definition of square roots: any number squared results in a positive product, whether the original number was positive or negative.

Thus, for 4900, while we see from perfect squares and factorization that 70 is the root, we must also consider its negative counterpart. This means that

• the positive square root is +70
• the negative square root is -70

These represent both solutions to the equation \( x^2 = 4900 \). By recognizing this principle, it becomes easier to solve quadratic equations and appreciate the nature of numbers in a wider mathematical context. Always remember: both positive and negative roots play a crucial role in real-world applications, from physics to finance.