Problem 454 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\sqrt[3]{x^{9}}\) (b) \(\sqrt[4]{y^{12}}\)

Step-by-Step Solution

Answer

(a) \(x^{3}\), (b) \(y^{3}\)

Step 1: Understand the Problem

The task is to simplify the given radicals: (a) \(\sqrt[3]{x^{9}}\) and (b) \(\sqrt[4]{y^{12}}\). Simplifying means rewriting them in the simplest form.

Step 2: Simplify \(\sqrt[3]{x^{9}}\)

To simplify \(\sqrt[3]{x^{9}}\), use the property of radicals that says \(\sqrt[n]{a^{m}} = a^{m/n}\). Hence, \(\sqrt[3]{x^{9}} = x^{9/3} = x^{3}\).

Step 3: Simplify \(\sqrt[4]{y^{12}}\)

Similarly, use the same property of radicals. \(\sqrt[4]{y^{12}}\) becomes \(y^{12/4} = y^{3}\).

Key Concepts

Radical Expressions

Radical expressions involve roots of numbers or variables. The most common radicals are square roots and cube roots.
For example, in \(\backslashsqrt[3]{x^{9}}\) (cube root of \(x^{9}\)), and \(\backslashsqrt[4]{y^{12}}\) (fourth root of \(y^{12}\)), we are dealing with cube roots and fourth roots respectively.
Simplifying radicals often involves using basic properties of radicals and exponents:

• Rule of Radicals: \(\sqrt[n]{a^{m}} = a^{m/n}\).
• The expression inside the radical is called the radicand.

Breaking down expressions into their simplest form helps in further algebraic manipulation and makes solving equations easier.

Properties of Exponents

Understanding the properties of exponents is crucial to simplifying radicals and other algebraic expressions. One key property is \(\sqrt[n]{a^{m}} = a^{m/n}\).
For example, consider \(\sqrt[3]{x^{9}}\). Using the rule, we can rewrite this as \(x^{9/3}\), which simplifies to \(x^{3}\).
Similarly, \(\sqrt[4]{y^{12}}\) becomes \(y^{12/4}\), which simplifies to \(y^{3}\).

Here are some essential properties of exponents to remember:
• Multiplication: \(a^{m} \cdot a^{n} = a^{m+n}\)
• Division: \(a^{m} / a^{n} = a^{m-n}\)
• Power of a power: \((a^{m})^{n} = a^{m \cdot n}\)

Understanding these properties allows for the simplification of more complex expressions in algebra.

Algebraic Simplification

Algebraic simplification involves rewriting expressions in their simplest form, making further calculations easier. Simplifying radicals is one aspect of algebraic simplification.
By applying the properties of exponents and radicals, expressions become more manageable.

Steps to simplify \(\backslashsqrt[3]{x^{9}}\) and \(\backslashsqrt[4]{y^{12}}\):
• Identify the type of root (cube root, fourth root, etc.)
• Use the rule \(\sqrt[n]{a^{m}} = a^{m/n}\) to convert the radical into an exponent form
• Simplify the exponent

The final answers are \(x^{3}\) and \(y^{3}\) respectively.
Simplifying expressions helps in solving equations, graphing functions, and performing other algebraic operations efficiently.