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The square root of a number is a value which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because when 3 is multiplied by 3 (3 x 3), the result is 9. The square root is denoted by the symbol \( \sqrt{\cdot} \).
To determine whether the square root of a number is rational or irrational, it is helpful to decide if the number is a perfect square — that is, the product of an integer multiplied by itself. In our exercise, \( \sqrt{100 a^{6}} \) includes the square root of 100, which is known to be 10, a rational number, because 10 x 10 = 100. However, we need to examine the square root of \( a^{6} \) as well to draw a conclusion about the entire expression.
When taking the square root of an exponent, the power is halved. So, the square root of \( a^{6} \) is \( a^{3} \), since 6 divided by 2 is 3. This simplification is only straightforward when the exponent is an even number. With these points in mind, we are now equipped to better understand whether the expression represents a rational or an irrational number.

A perfect square is an integer that's the square of an integer. In other words, it can be written as the product of some integer with itself. For instance, 16 is a perfect square because it can be expressed as 4 x 4. In the exercise, 100 is identified as a perfect square since it is 10 x 10.
Recognizing perfect squares is crucial because the square root of a perfect square is always an integer, and therefore a rational number. Integers are rational by definition since they can be expressed as the ratio of themselves to one. Non-perfect squares, on the other hand, often have square roots that are irrational numbers, like \( \sqrt{2} \) or \( \sqrt{3} \). These cannot be perfectly expressed as a ratio of two integers.
For effective problem-solving, especially in algebra, being able to distinguish between perfect squares and non-perfect squares is a foundational skill. This knowledge directly guided us in the solution to assess the rationality of \( \sqrt{100 a^{6}} \).

An exponent is a notation that represents how many times a number, called the base, is multiplied by itself. For example, \( a^{3} \) indicates that 'a' is used as a factor three times: \( a \times a \times a \). In our textbook exercise, the exponent is 6 in the term \( a^{6} \), signifying \( a \times a \times a \times a \times a \times a \).
It's important to understand that when an exponent is an even number, as in the case of 6, any positive base raised to that power will result in a perfect square. This is because the base will be multiplied by itself an even number of times. Hence, \( a^{6} \) is a perfect square regardless of the value of 'a', as long as 'a' is positive. This understanding is pivotal in resolving whether the square root of \( a^{6} \) is rational or not.
In the context of our problem, knowing how to work with exponents helped us simplify \( \sqrt{100 a^{6}} \) to a simpler expression of \( 10a^{3} \) by halving the exponent. Through this step, we’re able to recognize that the expression is indeed a rational number, as it involves the root of perfect squares only.