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An irrational number is a type of number that cannot be expressed as a simple fraction or ratio of two integers. These numbers have non-repeating and non-terminating decimal forms. Classic examples include numbers like \( \pi \), \( e \), and square roots of non-perfect squares like \( \sqrt{2} \) or \( \sqrt{10} \). It means we cannot find two integers, a and b, such that \( \frac{a}{b} \) represents their value exactly.

When dealing with fractions, an irrational number in the denominator creates a bit of a problem. Many math processes require or prefer the denominator to be a rational number. This is where the process of rationalizing the denominator, such as replacing \( \sqrt{10} \) with 10 as seen in the solution you are working on, becomes essential.

Simplifying fractions is the process of reducing the fraction to its simplest form, where the numerator and denominator have no common factors other than 1. This makes calculations and comparisons much easier.

In the given exercise, after multiplying \( \frac{2}{\sqrt{10}} \) by \( \sqrt{10} \), we get \( \frac{2\sqrt{10}}{10} \). The numerator and denominator share a common factor of 2, which allows further simplification. Therefore, we divide both by 2, simplifying the fraction to \( \frac{\sqrt{10}}{5} \). This is the simplest form where further reduction isn't possible since 5 and \( \sqrt{10} \) share no common factors.

• Finding the greatest common divisor (GCD) can help simplify fractions.
• Always aim for the smallest possible whole numbers in both the numerator and denominator.

Multiplying radicals involves combining like terms or roots in a way that follows the properties of exponents. When multiplying two same radicals, you essentially remove the radical by squaring. For example, \( \sqrt{a} \times \sqrt{a} = a \). This principle is crucial in rationalizing a denominator that contains irrational numbers.

In the exercise, we multiply \( \sqrt{10} \) by itself, following the identity \( \sqrt{10} \times \sqrt{10} = 10 \). This multiplication eliminates the radical, resulting in a simple integer in the denominator, hence making the fraction easier to handle and further simplify.

• Ensure that you are multiplying with the exact same radical to simplify effectively.
• Remember that multiplying different radicals requires using the product property \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \), which still holds true but doesn’t help with rationalizing.