Problem 529 - Elementary Algebra

Question

In the following exercises, write with a rational exponent. (a) \(\sqrt[8]{r}\) (b) \(\sqrt[12]{5}\) (c) \(\sqrt[4]{t}\)

Step-by-Step Solution

Answer

(a) \(r^{1/8}\), (b) \(5^{1/12}\), (c) \(t^{1/4}\)

Step 1: Understanding Radical Notation

When dealing with roots, the notation \(\root[n]{x}\) can be converted to rational exponents. The base of the root becomes the base of the exponent and the index becomes the denominator.

Step 2: Convert \(\root[8]{r}\) to a Rational Exponent

For part (a), \(\root[8]{r}\) can be written as \(r^{1/8}\). The denominator 8 is the index of the root.

Step 3: Convert \(\root[12]{5}\) to a Rational Exponent

For part (b), \(\root[12]{5}\) can be written as \(5^{1/12}\). The denominator 12 is the index of the root.

Step 4: Convert \(\root[4]{t}\) to a Rational Exponent

For part (c), \(\root[4]{t}\) can be written as \(t^{1/4}\). The denominator 4 is the index of the root.

Key Concepts

Understanding Radical Notation

Radical notation is a way to represent roots, such as square roots or cube roots. When you see an expression like \(\root[n]{x}\), it’s displaying that we want to find the n-th root of x. The symbol inside the root is called the radicand (in this case, x). The small number outside the radical sign is known as the index of the root (in this case, n).

If there is no number, it's understood to be a square root, meaning n=2. Understanding this notation helps to easily transition into rational exponents.

Recognizing the Index of the Root

The index of a root is the small number you see above and to the left of the radical sign, which tells you the degree of the root. For example, in \(\root[4]{t}\), the index is 4, indicating that this is a fourth root.

It's important to recognize this number because the root index becomes the denominator when converting to rational exponents. Thus, the index determines how many times a number must be multiplied by itself to return to the original number under the root.

Conversion to Rational Exponents

To convert a root into a rational exponent, you use the index as the denominator of a fraction. This fraction becomes the exponent of the base. For example:

- For \(\root[8]{r}\), write it as \(r^{1/8}\).
- For \(\root[12]{5}\), express it as \(5^{1/12}\).
- For \(\root[4]{t}\), convert it to \(t^{1/4}\).

This method allows us to use properties of exponents when working with roots, making calculations easier while preserving the original values.