Problem 74 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\frac{\sqrt[3]{4}}{\sqrt[3]{2}}\)

Step-by-Step Solution

Answer

\( \sqrt[3]{2} \) is the simplest radical form.

Step 1: Simplify the Expression

The given expression is \( \frac{\sqrt[3]{4}}{\sqrt[3]{2}} \). To simplify it, we need to divide the radicals. Notice that \( 4 = 2^2 \), therefore \( \sqrt[3]{4} = \sqrt[3]{2^2} \). So we can rewrite the expression as \( \frac{\sqrt[3]{2^2}}{\sqrt[3]{2}} \).

Step 2: Apply the Property of Radicals

Use the property of radicals \( \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} \) for the expression \( \frac{\sqrt[3]{2^2}}{\sqrt[3]{2}} \). This simplifies the expression to \( \sqrt[3]{\frac{2^2}{2}} \).

Step 3: Simplify Inside the Radical

Inside the radical, simplify \( \frac{2^2}{2} = 2 \). Thus, the expression becomes \( \sqrt[3]{2} \).

Step 4: Conclusion: Simplest Radical Form

The simplest radical form of the given expression \( \frac{\sqrt[3]{4}}{\sqrt[3]{2}} \) is \( \sqrt[3]{2} \).

Key Concepts

Radical Expressions

Radical expressions are mathematical expressions that contain a root symbol, such as a square root or cube root. The root symbol is used to indicate that a number is being raised to a fractional power. The most common roots are square roots and cube roots, but there can be fourth roots, fifth roots, and so on.

In the expression \( \sqrt[n]{a} \), \( n \) is called the index of the radical and \( a \) is the radicand. If there is no number indicated as the index, it's understood to be 2, meaning it's a square root. For cube roots, you will see \( \sqrt[3]{} \).

Simplifying radical expressions often involves simplifying what's inside the root first and then rewriting the expression in its simplest form. This sometimes means breaking down a number into its prime factors, as seen in our example, where we broke 4 down into 2 squared.

• Radicals represent roots and are equivalent to fractional exponents.
• To simplify radical expressions, use properties of exponents.
• Simplify the radicand (the number inside the radical) when possible.

Fractional Exponents

Fractional exponents are another way to represent radicals. They make many calculations easier, particularly when it comes to multiplying or dividing roots.

The expression \( a^{\frac{m}{n}} \) means that the number \( a \) is raised to the \( m \)th power, and then the \( n \)th root is taken. For example, \( a^{\frac{1}{2}} \) is equivalent to \( \sqrt{a} \), and \( a^{\frac{1}{3}} \) is equivalent to \( \sqrt[3]{a} \).

Using fractional exponents makes it easier to apply the laws of exponents, such as adding or subtracting exponents when multiplying or dividing like bases. This is particularly handy in algebra when simplifying expressions.

• The numerator of the fractional exponent is the power to which the base is raised.
• The denominator is the root that is taken.
• Fractional exponents allow for easy use of exponent rules to simplify expressions.

Properties of Radicals

The properties of radicals are essential tools when simplifying radical expressions. These properties help us manipulate and combine radicals in a structured way.

One important property is \( \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} \), which states that you can divide two radicals with the same index by dividing the radicands and placing the result under one radical. This property was used in the example to simplify \( \frac{\sqrt[3]{4}}{\sqrt[3]{2}} \) to \( \sqrt[3]{2} \).

Another property is the product rule, which says \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \). This means you can multiply the radicands and then find the root of the result.

• Combining radicals helps simplify expressions.
• Ensure the radicals have the same index before applying these properties.
• Always simplify the expression inside the radical first.