Problem 71 - Algebra 2 and Trigonometry

Question

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ \frac{5^{1} a^{\frac{2}{3}}}{4^{\frac{1}{3}}} $$

Step-by-Step Solution

Answer

\( \sqrt[3]{\frac{5a^2}{4}} \)

Step 1: Express the Fractional Exponents as Radicals

The expression given is \( \frac{5^{1} a^{\frac{2}{3}}}{4^{\frac{1}{3}}} \). We start by expressing the fractional exponents as radicals. Recall that \( x^{\frac{m}{n}} = \sqrt[n]{x^m} \). This means \( a^{\frac{2}{3}} = \sqrt[3]{a^2} \) and \( 4^{\frac{1}{3}} = \sqrt[3]{4} \).

Step 2: Substitute Radicals for Fractional Exponents

Now, substitute the radicals back into the expression. This gives us \( \frac{5 \cdot \sqrt[3]{a^2}}{\sqrt[3]{4}} \).

Step 3: Simplify the Radical Expression

Since both the numerator and the denominator contain cube roots, we can combine them. This is expressed as \( \sqrt[3]{\frac{5a^2}{4}} \) using the property \( \frac{\sqrt[n]{x}}{\sqrt[n]{y}} = \sqrt[n]{\frac{x}{y}} \).

Key Concepts

Fractional Exponents

Understanding fractional exponents is essential when dealing with radical expressions. These exponents contain a numerator and a denominator that denotes both power and root.
Fractional exponents of the form \( x^{\frac{m}{n}} \) can be translated into radical expressions. Here:

• The numerator "\( m \)" represents the power to which the number is raised.
• The denominator "\( n \)" indicates the type of root.

This means \( x^{\frac{m}{n}} = \sqrt[n]{x^m} \). For example, in the expression \( a^{\frac{2}{3}} \), "\( a \)" is squared, and then the cube root is taken.
Becoming comfortable with translating between fractional exponents and radicals can clarify many complex expressions and allow for easier manipulation and simplification of equations.

Cube Roots

A cube root is a special type of radical expression. It involves the root where the radicand is raised to the third power. For any positive number "\( x \)", the cube root of \( x \) is the number which, when multiplied by itself three times, gives \( x \).
Mathematically, this is expressed as \( \sqrt[3]{x} \).

• Cube roots are useful for simplifying expressions with exponents.
• They are particularly helpful when dealing with fractional exponents having "3" in the denominator, as in \( x^{\frac{2}{3}} \).

Remember, cube roots can turn complex-looking terms into simpler forms by reducing the exponents. This is seen in our expression transformation, moving from \( \frac{5 \cdot \sqrt[3]{a^2}}{\sqrt[3]{4}} \) to the combined cube root \( \sqrt[3]{\frac{5a^2}{4}} \). Understanding this will help simplify many challenging problems.

Simplifying Expressions

Simplifying expressions means making them as basic as possible without changing their value. With radical expressions, we strive to remove complex radicals in the numerator and denominator by combining or reducing similar terms.
For instance, in our example, we start with separate cube roots in the numerator and denominator. Using the property \( \frac{\sqrt[n]{x}}{\sqrt[n]{y}} = \sqrt[n]{\frac{x}{y}} \), we can merge them under a single radical:

• This step reduces complication, making the expression easier to interpret and calculate.
• The result is a streamlined expression with a single radical, \( \sqrt[3]{\frac{5a^2}{4}} \).

Mastering the art of simplification is crucial, as it transforms messy and difficult expressions into straightforward and understandable forms. It elevates mathematical efficiency and accuracy in solving problems.