Problem 80 - Intermediate Algebra

Question

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[9]{y^{6} z^{3}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( y^{\frac{2}{3}} z^{\frac{1}{3}} \).

Step 1: Express the Radical in Exponential Form

First, express \( \sqrt[9]{y^6 z^3} \) using fractional exponents. The 9th root of any expression can be represented as raising that expression to the power of \( \frac{1}{9} \). Thus, \( \sqrt[9]{y^6 z^3} = (y^6 z^3)^{\frac{1}{9}} \).

Step 2: Apply the Exponent Rule Individually

Apply the rule of exponents \((ab)^m = a^m b^m\) to separate the terms inside the parentheses: \[ (y^6 z^3)^{\frac{1}{9}} = y^{6 \cdot \frac{1}{9}} z^{3 \cdot \frac{1}{9}} \].

Step 3: Simplify Each Exponent

Now, simplify the exponents by performing the multiplication: \( y^{6 \cdot \frac{1}{9}} = y^{\frac{6}{9}} \) and \( z^{3 \cdot \frac{1}{9}} = z^{\frac{3}{9}} \). Simplify the fractions: \( \frac{6}{9} = \frac{2}{3} \) and \( \frac{3}{9} = \frac{1}{3} \).

Step 4: Write the Final Simplified Expression

Combine the simplified exponents: \( y^{\frac{2}{3}} z^{\frac{1}{3}} \). Therefore, \( \sqrt[9]{y^6 z^3} \) simplifies to \( y^{\frac{2}{3}} z^{\frac{1}{3}} \).

Key Concepts

Radical Expressions

Radical expressions involve roots of numbers or variables, and are often represented using the radical symbol \( \sqrt{} \). For example, \( \sqrt[n]{a} \) denotes the \( n \)-th root of \( a \). They express a repeated division process, much like how exponents are used for repeated multiplication. It’s important for these expressions to have positive radicands (the numbers under the root), especially when dealing with even roots like square roots or fourth roots. This ensures that the expression remains defined within the real number system.

Radical expressions can be rewritten using their equivalent expression with fractional exponents. For example, the square root \( \sqrt{a} \) can also be expressed as \( a^{1/2} \). This conversion is particularly useful when it comes to simplifying complex expressions or solving equations that involve radicals.

Exponent Rules

Understanding exponent rules is crucial when working with expressions involving powers, whether they have integer exponents or fractional exponents known as rational exponents. Here are some key exponent rules to keep in mind:

• Product of Powers Rule: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power Rule: \((a^m)^n = a^{m\cdot n}\)
• Power of a Product Rule: \((ab)^m = a^m \cdot b^m\)

In the context of the original problem, the Power of a Product Rule was used to break down the expression \((y^6 z^3)^{\frac{1}{9}}\). This rule allowed us to separate the powers of \(y\) and \(z\) individually, which simplified the process greatly.

Applying these rules can transform complex expressions into more manageable forms that are easier to work with or simplify, helping to streamline calculations in algebra.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that makes it easier to interpret, analyze, and solve mathematical problems. When simplifying expressions with exponents, the goal is to present the expression in its simplest form without changing its value.

In our example, the expression was initially presented as \( \sqrt[9]{y^6 z^3} \), a nested radical expression. The simplification process involved:

• Converting Radicals: The nested radical was rewritten as \((y^6 z^3)^{\frac{1}{9}}\) using rational exponents. This is useful because it allows easier application of exponent rules.
• Applying Exponent Rules: By utilizing the Power of a Product Rule, each variable, \(y\) and \(z\), was raised to its respective fractional exponent.
• Reducing Fractions: The exponents \(\frac{6}{9}\) and \(\frac{3}{9}\) were simplified to \(\frac{2}{3}\) and \(\frac{1}{3}\) respectively. This step simplifies the expression even further.

Ultimately, simplifying expressions helps in expressing data clearly and concisely, making further mathematical operations more straightforward. Therefore, learning how to apply rules correctly and reduce fractions efficiently is a vital part of algebraic study.