Problem 71 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\frac{\sqrt[3]{27}}{\sqrt[3]{4}}\)

Step-by-Step Solution

Answer

The simplest radical form is \( \frac{3}{\sqrt[3]{4}} \).

Step 1: Understanding the Problem

To simplify the expression \( \frac{\sqrt[3]{27}}{\sqrt[3]{4}} \), we need to express each cube root in simpler terms if possible. The goal is to simplify the entire radical expression into its simplest radical form.

Step 2: Simplify the Numerator

The numerator of the expression is \( \sqrt[3]{27} \). Since 27 is equal to \( 3^3 \), the cube root of 27 simplifies to 3, as \( \sqrt[3]{27} = 3 \).

Step 3: Rewrite the Denominator

The denominator is \( \sqrt[3]{4} \). Since 4 cannot be expressed as a perfect cube, it remains as it is.

Step 4: Combine and Simplify the Expression

After simplifying the numerator to 3, the expression becomes \( \frac{3}{\sqrt[3]{4}} \). This is the simplest form using radicals as \( \sqrt[3]{4} \) cannot be simplified further.

Key Concepts

Cube Roots

In mathematics, cube roots are a way to find a number that, when multiplied by itself three times, returns the original number. Specifically, the cube root of a number \(x\) is denoted as \(\sqrt[3]{x}\). If \(x\) can be expressed as \(a^3\), then the cube root is simply \(a\). This makes it easy to simplify cube roots if you can find the base that satisfies this equation.

For example, the cube root of 27 is calculated based on the fact that \(27 = 3^3\), so \(\sqrt[3]{27} = 3\). Cube roots, like any roots, are used to "undo" the process of raising a number to a power. Understanding cube roots is essential in simplifying expressions that involve radicals, especially when you need to simplify fractions or other expressions with radical components.

• Cube roots allow for simplification of perfect cubes easily.
• If a number is not a perfect cube, the expression stays under the cube root sign.

Numerators and Denominators

When working with fractions, it's important to understand the roles of the numerator and the denominator. The numerator is the top number in a fraction, representing how many parts of the whole are being considered. The denominator, on the other hand, is the bottom number, representing the total number of equal parts the whole is divided into.

In the context of the radical expression \(\frac{\sqrt[3]{27}}{\sqrt[3]{4}}\), \(\sqrt[3]{27}\) serves as the numerator, and \(\sqrt[3]{4}\) is the denominator. Simplifying these terms helps to make the fraction easier to work with, especially when performing operations or comparing fractions. Understanding how to handle numerators and denominators is critical in making complex expressions more manageable.

• Numerators tell you the size of the part you're dealing with.
• Denominators define how many parts make the whole.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and other higher roots. These expressions can often be simplified by identifying and extracting perfect powers. In the case of cube roots, we specifically look for numbers raised to the power of three.

Simplifying radical expressions often involves finding a number that fits neatly into the radical, enabling a quick simplification. For example, the cube root of 27 is simplified to 3 because 27 is a perfect cube. However, not all numbers simplify perfectly, as seen with 4 in \(\sqrt[3]{4}\).

• Radical expressions include roots such as square, cube, and more.
• Simplification depends on identifying perfect powers within those expressions.

Fraction Simplification

Simplifying fractions that contain radicals can be tricky. The goal is to rewrite the fraction in its simplest form by simplifying both the numerator and the denominator, if possible. In our example, \(\frac{\sqrt[3]{27}}{\sqrt[3]{4}}\), the numerator \(\sqrt[3]{27}\) simplifies to 3, resulting in \(\frac{3}{\sqrt[3]{4}}\).

It's critical to ensure that both components of the fraction are in their simplest form. This process may involve rationalizing the denominator if necessary, particularly if it contains surds or irrational numbers.

• Always simplify both the numerator and the denominator separately.
• Rationalizing might be needed for certain expressions to simplify their form further.