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A square root is a value that, when multiplied by itself, gives the original number. In other words, if you have a number like 144, its square root is 12 because 12 times 12 equals 144. The symbol for square root is \(\backslash\backslash\backslash\backslashsqrt{}\). Understanding square roots is important for simplifying radicals.

For instance, \(\backslash\backslash\backslash\backslashsqrt{144} = 12\) because 12 \(*\) 12 = 144. Similarly, \(\backslash\backslash\backslash\backslashsqrt{169} = 13\) because 13 \(*\) 13 = 169.

When you come across a fraction inside a square root, like \(\backslash\backslash\backslash\backslashsqrt{\frac{144}{169}}\), you can simplify it by separately taking the square roots of the numerator and the denominator.

Fraction simplification is the process of reducing a fraction to its simplest form. This means ensuring that the numerator and denominator have no common factors other than 1. For example, \(\frac{12\times\12}{13\times13}\) simplifies to \(\frac{12}{13}\).

In the context of simplifying radicals, when you have a fraction inside the square root, you handle the top (numerator) and bottom (denominator) parts separately.

So, if you start with \(\sqrt{\frac{144}{169}}\), you first find the square root of 144, which is 12, and the square root of 169, which is 13. The simplified form is \(\frac{12}{13}\).

By breaking down the components individually, simplifying radicals that contain fractions becomes straightforward and less daunting.

In any fraction, the numerator is the number on the top, and the denominator is the number on the bottom. These two components are crucial for understanding and solving fractions.

When simplifying, you often need to perform operations separately on the numerator and the denominator. For example, in \(\frac{144}{169}\), 144 is the numerator, and 169 is the denominator.

To simplify \(\sqrt{\frac{144}{169}}\), you take the square root of the numerator and the denominator separately. So, \( \sqrt{144} \) becomes 12 (numerator), and \( \sqrt{169} \) becomes 13 (denominator).

This gives you the fraction \( \frac{12}{13} \). Understanding the roles of numerator and denominator makes fraction simplification more manageable.