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Understanding the difference between rational and irrational numbers is foundational in mathematics. Rational numbers are numbers that can be expressed as a fraction or ratio of two integers, where the numerator is an integer, and the denominator is a non-zero integer. Examples include \frac{1}{2}, \frac{4}{3}, and -\frac{7}{5}. They include integers, as every integer can be written as a fraction with a denominator of 1.

On the other hand, irrational numbers cannot be expressed as a simple fraction. They have decimal expansions that neither terminate nor become periodic. The most famous examples are \(\pi\) and \(\sqrt{2}\), but there are many others, such as the cube root of any non-perfect cube number, \(\sqrt[3]{5}\), or Euler's number, \(e\). Understanding that \(\sqrt{2}\) is irrational is crucial to solving our cube root problem, as it leads to the realization that if \(\sqrt{2}\) itself is irrational, its cube would also share this property.

Proof by contradiction is a powerful method of mathematical proof. In essence, this technique involves assuming that the opposite of what you want to prove is true. Then, through a series of logical deductions, you arrive at an inconsistency or a statement that is known to be false. This contradiction shows that the original assumption must be false, thereby proving that the original statement is true.

In the context of our exercise with \(\sqrt{2}\)^3, we begin by assuming this expression is a rational number. By manipulating this assumption and eventually finding that it leads us to the conclusion that \(\sqrt{2}\) is rational (which is false), we've stumbled upon our contradiction. This conflict within our logic is the key indicator that our initial assumption is incorrect, thus affirming that \(\sqrt{2}\)^3 is indeed irrational.

A cube root of a number is a value that, when multiplied by itself three times (cubed), gives the original number. The cube root of x is denoted as \(\sqrt[3]{x}\). Unlike square roots, which can have both a positive and a negative root (since squaring either value will give the positive original number), cube roots will retain the sign of the original number.

For the number 2, its cube root is denoted as \(\sqrt[3]{2}\), and it represents a number that, when used in multiplication three times, equals 2. To connect this concept to a practical example, if one has a volume of a cube that is 2 cubic units, the length of each side of this cube would be the cube root of 2.

It's important to note that the cube roots of perfect cubes (like 8 or 27) are rational, while the cube roots of non-perfect cubes (like 2 or 5) are irrational. In the case of \(\sqrt[3]{2}\), this extends to \(\sqrt[2}\)^3; since the cube root is irrational, when it is cubed, it remains irrational. This pertains to our exercise and the reason why we can confidently say that \(\sqrt{2}\)^3 is an irrational number.