Problem 37 - Algebra for College Students

Question

Write each of the following in radical form. For example, \(3 x^{\frac{2}{3}}=3 \sqrt[3]{x^{2}}\). \((2 x-3 y)^{\frac{1}{2}}\)

Step-by-Step Solution

Answer

\((2x - 3y)^{\frac{1}{2}} = \sqrt{2x - 3y}\).

Step 1: Identify the Expression

The expression given is \((2x - 3y)^{\frac{1}{2}}\). This indicates the expression is raised to the power of \(\frac{1}{2}\).

Step 2: Recognize the Form of Radical Expressions

Expressions of the form \(a^{\frac{m}{n}}\) can be rewritten in radical form as \(\sqrt[n]{a^{m}}\). The denominator \(n\) of the fraction becomes the root, and the numerator \(m\) remains the power inside.

Step 3: Apply the Radical Form Conversion

For \((2x - 3y)^{\frac{1}{2}}\), let \(a = 2x - 3y\), \(m = 1\), and \(n = 2\). Substituting these values into the radical form, the expression becomes \(\sqrt{2x - 3y}\).

Step 4: Write the Final Radical Form

The final radical form of the expression \((2x - 3y)^{\frac{1}{2}}\) is \(\sqrt{2x - 3y}\).

Key Concepts

Exponent Rules

Exponent rules are the foundation of working with powers in mathematics. These rules tell us how to simplify and manipulate expressions involving exponents. Here are the basic rules:

• Whenever you multiply like bases, you add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• For division, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• An exponent of zero means the number is equal to one: \(a^0 = 1\) if \(a eq 0\).
• Raising a power to another power means you multiply the exponents: \((a^m)^n = a^{m \cdot n}\).

Whenever we encounter a fraction as an exponent, such as \(\frac{1}{2}\), it indicates a root. The denominator tells us the type of root, and the numerator stays as the power inside the radical. For example, \(a^{\frac{1}{2}}\) becomes \(\sqrt{a}\). This is an important rule to remember when converting expressions into radical forms.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve equations or to write them in a desired form. This process includes combining like terms, factoring, and applying operation rules like the distributive property.
To convert expressions with fractional exponents to radical forms, recognizing patterns in Algebra is key.

• Identify the base and the exponent of the expression.
• Match the numerator to the power inside the radical and the denominator to the root.
• Rearrange terms if needed, to make simplification easier.

In the expression \((2x - 3y)^{\frac{1}{2}}\), we treat \(2x - 3y\) as a single entity inside the radical formed by the exponent \(\frac{1}{2}\). This approach simplifies working with more complex components of an expression without distributing or expanding unnecessarily, keeping the expression manageable and clear.

Mathematical Expressions

Expressions in mathematics consist of numbers, variables, and mathematical operators such as addition, subtraction, multiplication, and division. Understanding how to read and convert these expressions is crucial.

• A mathematical expression can represent real-world problems or abstract concepts.
• Algebraic expressions vary in complexity from simple numbers to those involving operations and variables.
• Expressions like \(\sqrt{2x - 3y}\) show radical usage where an entire expression is encompassed into a single radical term.

Mathematical expressions with radical components can sometimes seem challenging.
However, by simplifying and manipulating expressions into familiar forms, they become more accessible. For instance, the expression \((2x - 3y)^{\frac{1}{2}}\) was rewritten as \(\sqrt{2x - 3y}\), improving readability and understanding while maintaining its value. Converting such expressions requires understanding the structure and properties of radicals and exponents. It's about recognizing the symmetry and relationships between different parts of the expression, making radical expressions not only simpler but more elegant.