Problem 102 - College Algebra

Question

In Exercises \(95-102,\) simplify by reducing the index of the radical. $$\sqrt[12]{x^{4} y^{8}}$$

Step-by-Step Solution

Answer

The simplified form of the expression \( \sqrt[12]{x^{4} y^{8}} \) is \( x^{1/3}y^{2/3} \)

Step 1: Identify the base and exponent

Identify the bases and their exponents in the given expression. Here, the bases are x and y with exponents 4 and 8 respectively. The index of the radical is 12.

Step 2: Apply the exponent rule

Apply the exponent rule for radicals. The rule is that a base raised to an exponent, all under a root, can be simplified by dividing the exponent by the index of the root.\( \sqrt[12]{x^{4}} = x^{4/12} = x^{1/3} \) and \( \sqrt[12]{y^{8}} = y^{8/12} = y^{2/3}\).

Step 3: Simplify the radical

Simplify the radical by combining the simplified expressions.\( \sqrt[12]{x^{4} y^{8}} = x^{1/3}y^{2/3} \)

Key Concepts

Exponent Rules

Understanding how to work with exponents is essential in simplifying expressions involving radicals. Exponents indicate how many times a number, known as the base, is multiplied by itself. In expressions like \(x^4\), the 4 is the exponent and x is the base. When you apply the exponent rules, each variable's exponent in a term can be simplified. One of the most useful rules is when dealing with radicals, the use of fractional exponents.

• For any base \(a\) with a radical \(\sqrt[n]{a^m}\), it can be rewritten using fractional exponents as \(a^{m/n}\).
• The major advantage of using this conversion is that it allows simplifying expressions more easily by performing arithmetic operations on the exponents.

Remember, exponent rules help in performing operations like multiplication and division on terms with the same base, by adding or subtracting the exponents respectively.
Converting radicals to fractional exponents is particularly handy in algebra because it unifies different forms of expressions into a single format.

Roots and Indices

Roots and indices form the backbone of understanding radical expressions. The concept of a root deals with finding the value that can be repeatedly multiplied to obtain a certain number. The index of a radical indicates the degree of the root.

• For example, \(\sqrt[12]{x^4}\) means the 12th root of \(x^4\), where 12 is the index and 4 is the exponent of the base \(x\).
• To simplify, divide the exponent by the index, which helps in reducing the expression under the radical.

The same concept applies to other bases and their respective exponents.
Simplifying involves ensuring all indices are used effectively to reduce the expression while maintaining the original value. Therefore, when you have an expression like \(\sqrt[n]{a^b}\), remember to simplify it using \(a^{b/n}\).
Roots are pivotal in converting complex expressions into more manageable forms, ensuring equations are easier to handle.

Algebraic Expressions

Algebraic expressions are combinations of variables, exponents, and constants that make up the mathematical sentences we work with. Understanding how to simplify them is crucial for solving equations or manipulating expressions.
When handling expressions involving radicals and exponents, clarity in the algebraic manipulation process is key:

• Identify the base and the exponents first. This will guide the rest of the simplification process.
• Use exponent rules to handle the indices and roots.

By applying the fractional exponent rule, expressions with large indices can be broken down.
Simplification often leads to expressions being easier to compute and compare. In the given exercise, once each term is simplified, combining them together leads to the final result \(x^{1/3}y^{2/3}\).
The skill of simplifying algebraic expressions is foundational and helps in various areas of mathematics and applications in real-world problems. As you grow more familiar with these transformations, handling complex expressions will become a seamless process. Keep practicing these to bolster your algebraic problem-solving skills.