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Square roots are an important concept in mathematics. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by itself (4 × 4) equals 16.

This idea is often represented using the radical symbol \(\sqrt{}\). In the case of \(\sqrt{\frac{1}{16}}\), we are looking for a number that, when multiplied by itself, gives the fraction \(\frac{1}{16}\).

We can break this down using the property of square roots for fractions: \[\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\]. This means we can take the square root of the numerator (top part) and the denominator (bottom part) separately.

Simplifying fractions makes them easier to understand and work with. A fraction is simplified when the numerator and the denominator have no common factors other than 1. For example, in the fraction \(\frac{1}{16}\), the numerator is 1 and the denominator is 16, which have no common factors.

Applying the property of square roots to this fraction, we get \( \sqrt{\frac{1}{16}} = \frac{\sqrt{1}}{\sqrt{16}}\).

Separate the square root of each part: \(\sqrt{1} = 1\) because 1 × 1 = 1 and \(\sqrt{16} = 4\) because 4 × 4 = 16. So, \[ \frac{\sqrt{1}}{\sqrt{16}} = \frac{1}{4}\]. Hence, \(\sqrt{\frac{1}{16}} = \frac{1}{4}\).

Basic algebra involves the use of variables and constants to represent numbers and perform operations. In our example, we used a property of square roots to simplify the expression \(\sqrt{\frac{1}{16}}\).

By applying algebraic principles, we were able to break the problem into smaller, manageable steps. These smaller steps included identifying the expression, applying the square root properties, and simplifying the numerator and denominator separately.

Understanding these basic algebraic tools allows us to solve more complex problems and forms the foundation for higher-level mathematics. With practice, concepts like square roots and simplifying fractions become easier and more intuitive.