Problem 68 - Algebra for College Students

Question

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(9 x^{2} y^{4}\right)^{\frac{1}{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(3xy^2\).

Step 1: Identify the Problem

We need to simplify the expression \( \left(9x^2 y^4\right)^{\frac{1}{2}} \) by breaking down the components and expressing the final result using positive exponents.

Step 2: Apply the Exponent Rule

According to the exponent rule, \((a^m)^n = a^{m \cdot n}\), apply \(\frac{1}{2}\) as the exponent to each component within the parentheses: \(9^{\frac{1}{2}}\), \(x^{2 \cdot \frac{1}{2}}\), and \(y^{4 \cdot \frac{1}{2}}\).

Step 3: Simplify Each Component

Calculate each result: \(9^{\frac{1}{2}}\) simplifies to \(3\), \(x^{2 \cdot \frac{1}{2}}\) becomes \(x^{1}\) (because \(2 \cdot \frac{1}{2} = 1\)), and \(y^{4 \cdot \frac{1}{2}}\) becomes \(y^{2}\) (since \(4 \cdot \frac{1}{2} = 2\)).

Step 4: Combine Simplified Components

Integrate the simplified parts into one expression: \(3xy^2\). This final expression contains only positive exponents as required.

Key Concepts

Exponent Rules

Exponent rules are essential tools in algebra that help manage and simplify expressions involving powers. Understanding these rules is crucial for effectively manipulating and solving equations. One of the most fundamental rules is the power of a power rule,

• \((a^m)^n = a^{m \times n}\)

This rule means when raising a power to another power, multiply the exponents. For example, given an expression like \((x^2)^3\), we apply the rule: multiply 2 and 3 to get \(x^{6}\).
Another important rule is the product of powers rule:

• \(a^m \times a^n = a^{m + n}\)

This rule dictates that when multiplying powers with the same base, you add the exponents. If you have \(x^2 \times x^3\), this simplifies to \(x^{5}\). These rules form the foundational building blocks for more complex algebraic operations.

Positive Exponents

Working with positive exponents means expressing all terms with non-negative exponents. Negative exponents can make expressions appear more complex, requiring additional steps to simplify. When simplifying expressions, it's essential to convert any negative exponents into positive ones for clarity and standard practice.
Positive exponents follow these basic rules:

• A positive exponent indicates how many times a base is multiplied by itself.
• If a base is raised to the power of 0, it equals 1: \(a^0 = 1\).
• To convert a negative exponent, place the term in the denominator: \(a^{-n} = \frac{1}{a^n}\).

These guidelines help ensure all components are in their simplest form.

Simplifying Expressions

The process of simplifying expressions aims to make complex algebraic expressions easier to understand and work with, often by reducing them to their most basic form. Simplification generally involves:

• Identifying like terms and combining them where possible.
• Applying exponent rules rigorously to combine and reduce terms.
• Rewriting the expression using only positive exponents, ensuring clarity and applicability.

Consider our exercise: \( \left(9x^2 y^4\right)^{\frac{1}{2}} \). By applying exponent rules, each component within the parentheses is addressed: \(9^{\frac{1}{2}}, x^{1}, y^{2}\). All are reduced to simplest terms ensuring the final expression \(3xy^2\) uses only positive exponents. Simplification makes the expression more adaptable for further computation or evaluation.