Problem 443 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\sqrt[3]{27}\) (b) \(\sqrt[4]{16}\) (c) \(\sqrt[5]{243}\)

Step-by-Step Solution

Answer

(a) 3; (b) 2; (c) 3

Step 1 - Understand the Problem

In each part of the exercise, you are asked to simplify a radical expression. This involves finding the number which, when raised to the indicated root, yields the number under the radical.

Step 2 - Simplify \(\root[3]{27} \)

Recognize that 27 can be expressed as a power: \(\root[3]{27} = \root[3]{3^3}\). Since the cube root of \(\root[3]{3^3}\) simplifies to 3, \(\root[3]{27} = 3\).

Step 3 - Simplify \(\root[4]{16} \)

Identify that 16 can be written as a power: \(\root[4]{16} = \root[4]{2^4}\). The fourth root of \(\root[4]{2^4}\) simplifies to 2, so \(\root[4]{16} = 2\).

Step 4 - Simplify \(\root[5]{243} \)

Notice that 243 can be expressed as a power: \(\root[5]{243} = \root[5]{3^5}\). The fifth root of \(\root[5]{3^5}\) simplifies to 3, making \(\root[5]{243} = 3\).

Key Concepts

nth roots

An nth root of a number is a value that, when raised to the power of n, gives the original number. Essentially, it’s the inverse operation of raising a number to a power. For instance, the cube root (3rd root) of 27 is 3 because 3 raised to the power of 3 equals 27, or \(3^3 = 27\). Similarly, the fourth root of 16 is 2 because \(2^4 = 16\). To simplify an nth root:

• Identify the number under the radical (the radicand).
• Express the radicand as an exponent involving a base number to the power of n.
• Simplify by identifying the base number that fits this criterion.

For example, to find the cube root of 27, express 27 as \(3^3\), and simplify to the base, which is 3.

exponents

Exponents are a way to express repeated multiplication of the same number. For example, \(2^4\) means multiplying 2 by itself 4 times. Exponents make it easier to write and work with very large or very small numbers. Here’s how you can handle them in the context of nth roots and radicals:

• Break down the radicand to its base and exponent.
• This helps identify the underlying structure, whether it’s a third root (cube root), fourth root, etc.
• Remember the basic properties of exponents, like \(a^m \times a^n = a^{m+n}\).

For example, simplifying \(\sqrt[5]{243} \) involves expressing 243 as \(3^5\), then recognizing that the fifth root of \(3^5\) is just 3.

algebraic simplification

Algebraic simplification involves reducing an expression to its simplest form. When simplifying radical expressions, follow these steps:

• Identify if the number under the radical (radicand) can be expressed as a power (e.g., \(3^3\) or \(2^4\)).
• Use the radical’s index to determine how to simplify (e.g., cube root, 4th root).
• Simplify the expression by reducing it to the base number.

Let’s illustrate this with an example: If you have \(\sqrt[4]{16}\), recognize that 16 is \(2^4\) and the 4th root of \(2^4\) is 2. Therefore, \(\sqrt[4]{16} = 2\).
This method ensures you systematically simplify complex radical expressions.