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When you see a radical expression like \(\root{n}{a^m}\), you can convert it to a form that uses exponents. This is known as expressing the radical as a fractional exponent. In general, \(\root{n}{a^m}\) can be written as \(a^{\frac{m}{n}}\). This makes it easier to work with because it follows the same exponent rules we use with whole numbers.
For example, \(\sqrt{a}\) is the same as \(a^{\frac{1}{2}}\), and \(\sqrt[3]{a^2}\) becomes \(a^{\frac{2}{3}}\). In our exercise, \(\sqrt[12]{x^{38}}\) is transformed into the fractional exponent form \(x^{\frac{38}{12}}\). This makes subsequent steps much simpler.

After converting to a fractional exponent, the next step is simplification. Simplifying involves making the expression as clean and small as possible.
This often involves reducing fractions: \(\frac{m}{n}\) inside the exponent. To do this, find the greatest common divisor (or GCD) of the numerator and denominator. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.
In our example, the fractional exponent is \(\frac{38}{12}\). To simplify this, we find the GCD of 38 and 12, which is 2. By dividing both the numerator and the denominator by 2, we get \(\frac{38 \div 2}{12 \div 2} = \frac{19}{6}\).
This reduces the exponent, making our original expression cleaner: \(\sqrt[12]{x^{38}}\) becomes \(x^{\frac{19}{6}}\).

The greatest common divisor (GCD) is a key concept in simplifying fractions and, by extension, exponents. It is the largest positive integer that divides two or more integers without leaving a remainder. To find the GCD of two numbers:

• List out the factors of each number.
• Identify the common factors.
• Choose the largest of these common factors.

For example: the factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 38 are 1, 2, 19, and 38. Their common factors are just 1 and 2, so the GCD is 2.

Once you have the GCD, you can simplify a fraction by dividing both the numerator and the denominator by the GCD. This process can also be applied to simplifying the exponents in radical expressions.