Problem 46 - Algebra for College Students

Question

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(\sqrt{2 x y}\)

Step-by-Step Solution

Answer

\(2^{\frac{1}{2}} x^{\frac{1}{2}} y^{\frac{1}{2}}\)

Step 1: Identify the Expression

The expression given is \(\sqrt{2xy}\). This is the square root of the product of 2, \(x\), and \(y\).

Step 2: Express Each Component Using Exponent

The square root, \(\sqrt{}\), can be rewritten using exponent notation as raising the expression inside the root to the power of \(\frac{1}{2}\).

Step 3: Apply the Exponent to Each Factor

Rewrite the square root expression as \((2xy)^{\frac{1}{2}}\). Then, distribute the exponent \(\frac{1}{2}\) to each factor: \(2^{\frac{1}{2}}\), \(x^{\frac{1}{2}}\), and \(y^{\frac{1}{2}}\).

Step 4: Combine into a Single Expression

Combine all the terms with positive exponents to form the final expression: \(2^{\frac{1}{2}} x^{\frac{1}{2}} y^{\frac{1}{2}}\).

Key Concepts

Square Root Transformation

The concept of square root transformation is key in simplifying algebraic expressions. When you see a square root, like \( \sqrt{2xy} \), you can transform it using exponents. The square root is equivalent to raising a number to the power of \( \frac{1}{2} \). This transformation is fundamental because it allows us to apply algebraic rules more easily.

Imagine you have a square root involving multiple variables or numbers, this can be expressed as one half power for each element inside the root. For example, \((2xy)^{\frac{1}{2}}\) means the same as \((2^{\frac{1}{2}})(x^{\frac{1}{2}})(y^{\frac{1}{2}})\). This makes calculations using exponents more systematic and uniform.

Exponent Rules

Exponent rules are foundational principles in algebra that enable us to manipulate exponents easily. Understanding these rules allows us to transform and simplify expressions effectively. One critical rule is the power of a product rule, which states that \( (ab)^n = a^n \cdot b^n \).

This is what we use when we distribute exponents across terms in an expression. For instance, in our example of \( (2xy)^{\frac{1}{2}} \), each component is affected by the same exponent separately: \( 2^{\frac{1}{2}}, x^{\frac{1}{2}}, \) and \( y^{\frac{1}{2}} \).

• Product of Powers: When multiplying like bases, add the exponents: \( a^m \cdot a^n = a^{m+n} \).
• Power of a Power: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).

Learning to leverage these rules can significantly simplify complex algebraic tasks.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. These expressions can become quite complex, but understanding their structure is crucial for manipulation and simplification.

An expression like \( 2^{\frac{1}{2}} x^{\frac{1}{2}} y^{\frac{1}{2}} \) involves numbers and variables combined using rules of exponents. It's important to recognize:

• Each variable or number is raised to an exponent.
• The operation between these terms is multiplication.
• The whole expression represents a product where each component contributes to the result.

By breaking down expressions into components with known rules (like exponentiation), we can bring order to manipulation and gain insight into the expression's meaning and applications.