Problem 51 - Algebra for College Students

Question

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(\sqrt[4]{a^{2} b^{3}}\)

Step-by-Step Solution

Answer

\( a^{\frac{1}{2}} b^{\frac{3}{4}} \)

Step 1: Convert the Entire Radical to an Exponent

The given expression is \( \sqrt[4]{a^2 b^3} \). The expression \( \sqrt[4]{x} \) can be rewritten as \( x^{\frac{1}{4}} \). Therefore, \( \sqrt[4]{a^2 b^3} \) can be written as \( (a^2 b^3)^{\frac{1}{4}} \).

Step 2: Apply Exponent to Each Base Inside the Radical

Apply the exponent \( \frac{1}{4} \) to each base inside the parentheses. For the expression \( (a^2 b^3)^{\frac{1}{4}} \), apply the exponent to both \( a^2 \) and \( b^3 \): \( a^{2 \cdot \frac{1}{4}} \) and \( b^{3 \cdot \frac{1}{4}} \).

Step 3: Simplify Each Part Separately

Multiply the exponents inside the parentheses. This gives us: \( a^{\frac{2}{4}} \) and \( b^{\frac{3}{4}} \). Simplify the exponent for \( a \): \( \frac{2}{4} = \frac{1}{2} \). This gives us \( a^{\frac{1}{2}} \). The exponent for \( b \) remains as it is: \( b^{\frac{3}{4}} \).

Step 4: Write the Final Answer

Combine both parts to write the entire expression with positive rational exponents. This gives you: \( a^{\frac{1}{2}} b^{\frac{3}{4}} \).

Key Concepts

Rational Exponents

Rational exponents are a way to express roots in the form of fractional powers. Instead of using the radial symbol to denote roots, we use fractions in the exponent to simplify expressions. For instance, a square root, often denoted as \(\sqrt{x}\), can be expressed as \(x^{\frac{1}{2}}\). This idea extends to any root. So, the fourth root \(\sqrt[4]{x}\) can be expressed with a rational exponent as \(x^{\frac{1}{4}}\). This representation not only simplifies the notation but also makes it easier to perform algebraic operations.
\(\textbf{Examples:}\)

• Cube root: \(\sqrt[3]{x} = x^{\frac{1}{3}}\)
• Fifth root: \(\sqrt[5]{x} = x^{\frac{1}{5}}\)

The use of rational exponents is essential in calculus and advanced mathematics because it streamlines the properties of exponents, making it simpler to multiply and divide root expressions.

Radical Expressions

Radical expressions are expressions that contain a radical sign, \(\sqrt{}\), which signifies roots, such as square roots, cube roots, and so on. They are common in algebra and calculus, often needing to be simplified for ease of use and for solving equations.
A radical expression can often be converted into one that uses rational exponents, as seen in the previous section. For example, converting \(\sqrt[4]{a^2 b^3}\) into \((a^2 b^3)^{\frac{1}{4}}\) allows us to use the properties of exponents to simplify it further.
The process involves the following steps:

• Identifying the radicand (the expression under the radical symbol): In our example, it's \(a^2 b^3\).
• Determining the type of root: This could be square root, cube root, fourth root, etc.
• Converting the radical to a fractional exponent: Such as \(\sqrt{a} = a^{\frac{1}{2}}\) or \(\sqrt[4]{x} = x^{\frac{1}{4}}\).

Translating radicals into expressions with rational exponents provides a versatile framework for integrating these expressions into equations and functions.

Simplifying Exponents

Simplifying expressions with exponents makes it easier to work with complex algebraic equations. When handling expressions like \((a^2 b^3)^{\frac{1}{4}}\), the process involves applying the fractional exponent to each term inside the parentheses and simplifying each separately.
To simplify:

• Distribute the exponent: Apply the outside exponent to each factor inside, transforming it into \(a^{2 \cdot \frac{1}{4}}\) and \(b^{3 \cdot \frac{1}{4}}\).
• Perform operations on each base: Calculate the new exponents, such as \(a^{\frac{2}{4}} = a^{\frac{1}{2}}\).
• Ensure all exponents are reduced: Simplify fractions like \(\frac{2}{4}\) to get \(\frac{1}{2}\).

Combining the simplified bases gives you \(a^{\frac{1}{2}} b^{\frac{3}{4}}\). This approach not only eases the complexity of the expression but lays down a clearer path for further manipulation in algebraic equations or calculus operations.