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nth root. Rational Exponents.

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The nth degree root

The nth degree root of a number can be found using the following formula: \( \sqrt[n]{a} = a^\frac{1}{n} \) Where "\(a\)" is the number to be rooted, and "\(n\)" is the degree of the root.
For instance, the fourth degree root of 81 can be found using the formula above:
\( \sqrt[4]{81}=81^\frac{1}{4} \)
We can simplify the expression above by using the fact that 81 is equal to 3 raised to the fourth power:
\( \sqrt[4]{81}=(3^4)^\frac{1}{4}=3^{4\cdot \frac{1}{4}}=3^1=3 \)
Therefore, the fourth degree root of 81 is 3.

It's worth noting that some nth degree roots can be irrational, meaning they cannot be expressed as a ratio of two integers. For example, the square root of \(2 (\sqrt{2}) \) is an irrational number because it cannot be written as a fraction.

Properties of nth Roots:

These properties and formulas of nth root are useful for simplifying and solving problems involving radicals.

Rational exponents

Rational exponents, also known as fractional exponents, are a way of representing powers and roots of a number in a more general and flexible way than using only integer exponents. A rational exponent is a number that can be expressed as a fraction, where the numerator represents the power to which the base is raised, and the denominator represents the root that is taken.

For example, let \(a\) be a positive real number and \(m\) and \(n\) be positive integers. Then the following are some examples of rational exponents:

\( a^\frac{1}{2}=\sqrt{a} \)

\( a^\frac{2}{3}=\sqrt[3]{a^2} \)

\( a^\frac{3}{4}=\sqrt[4]{a^3} \)

\( a^\frac{5}{2}=\sqrt{a^5} \)

In general, we can define a rational exponent as follows: \( a^\frac{m}{n} = \sqrt[n]{a^m} \), where \(a\) is a positive real number, \(m\) is an integer, and \(n\) is a positive integer.

Properties of exponents:

These properties of exponents allow us to simplify complex expressions and perform operations on them more easily. They are important not only in algebra, but also in many other fields of mathematics and science.