Problem 53 - Algebra for College Students

Question

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(\sqrt[5]{(2 x-y)^{3}}\)

Step-by-Step Solution

Answer

\((2x-y)^{\frac{3}{5}}\)

Step 1: Identify the Radical

The expression given is \( \sqrt[5]{(2x-y)^3} \). This is a radical expression where the radicand is \((2x-y)^3\) and the index of the root is 5.

Step 2: Apply the Rational Exponent Rule

According to the property of exponents, \( \sqrt[n]{a} = a^{\frac{1}{n}} \). Here, the index is 5, so apply it to the entire expression under the radical: \( (2x-y)^3 \) becomes \( (2x-y)^{\frac{3}{5}} \).

Step 3: Write the Final Expression

Using the above conversion, the expression \( \sqrt[5]{(2x-y)^3} \) can be rewritten as \( (2x-y)^{\frac{3}{5}} \) using positive rational exponents.

Key Concepts

Radical Expressions

Radical expressions are mathematical expressions that include a root symbol, often referred to as a "radical." When dealing with radicals, it is crucial to identify two main parts: the index of the root and the radicand. The radicand is the number or expression inside the radical sign, and the index is the small number written just outside and above the radical sign that tells you which root to take. For example, in the expression \( \sqrt[5]{(2x-y)^3} \), the radicand is \((2x-y)^3\) and the index is 5.
Radical expressions are often simplified using the properties of exponents, which help convert them into expressions with rational exponents. This conversion makes it easier to perform operations like multiplication, division, and finding the power of an expression. Understanding these conversions is essential for solving complex algebraic equations effectively.

Properties of Exponents

The properties of exponents allow us to manipulate expressions involving powers and roots systematically. Here are some fundamental properties to remember:

• \( a^{m} \times a^{n} = a^{m+n} \) – Multiplying powers with the same base
• \( \left(a^{m}\right)^{n} = a^{m \times n} \) – Raising a power to another power
• \( \sqrt[n]{a} = a^{\frac{1}{n}} \) – Converting roots to rational exponents

In the example \( \sqrt[5]{(2x-y)^3} \), we apply these properties. Specifically, the rule \( \sqrt[n]{a} = a^{\frac{1}{n}} \) converts the radical expression into one with a rational exponent: \( (2x-y)^{\frac{3}{5}} \). This simplifies calculations by transforming roots into exponents, allowing further simplification or manipulation using other properties.

Index of Roots

The index of a root is fundamental in understanding how to work with radical expressions. The index tells you the degree of the root you are taking. For instance, a square root implies an index of 2, while a cube root has an index of 3. In mathematical notation, roots are expressed as \( \sqrt[n]{a} \), where \( n \) is the index.
An important aspect of the index is its role in converting radical expressions into expressions with rational exponents. The relationship can be defined by the property that \( \sqrt[n]{a} = a^{\frac{1}{n}} \). This property is vital for simplifying expressions, especially when the calculations involve complex or high-degree radicals.
By understanding the index’s role, you can effectively translate any radical expression into one that uses rational exponents, making calculations more straightforward and manageable. For the expression \( \sqrt[5]{(2x-y)^3} \), the index 5 allows the conversion to a simpler form: \( (2x-y)^{\frac{3}{5}} \).