Problem 71 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Write the expression in simplest radical form. $$ \sqrt[3]{\frac{2}{3}} $$

Step-by-Step Solution

Answer

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

Step 1: Find the cube root of the numerator

In this case, the numerator is 2. Since 2 doesn't have an integer cube root, we'll leave it as \(\sqrt[3]{2}\).

Step 2: Find the cube root of the denominator

In this case, the denominator is 3. Since 3 doesn't have an integer cube root either, we'll leave it as \(\sqrt[3]{3}\).

Step 3: Simplify the fraction

Now we'll write the expression as a fraction of the cube roots. So, we have \(\frac{\sqrt[3]{2}}{\sqrt[3]{3}}\).

Step 4: Combine the cube roots

We can combine the cube roots by using the rule \(\frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}}\). Therefore, our final simplified radical expression is: $$ \sqrt[3]{\frac{2}{3}} $$ This is the simplest radical form of the given expression, as neither 2 nor 3 can be further simplified with cube roots.

Key Concepts

Cube Roots

A cube root is a number that, when multiplied by itself three times, equals a given number. In other words, finding the cube root of a number means determining what number can be cubed, or raised to the power of three, to result in the original number. The cube root is denoted by the radical symbol with an index of three, like this: \( \sqrt[3]{x} \).
For example, the cube root of 8 is 2, since \( 2 \times 2 \times 2 = 8 \). However, for numbers like 2 or 3, these do not have integer cube roots, leading us to leave the terms in their cube root form, \( \sqrt[3]{2} \) and \( \sqrt[3]{3} \).

• Cube roots are the opposite of cubing a number.
• They involve both positive and negative values (e.g., \( \sqrt[3]{-8} = -2 \)).
• Only perfect cubes have integer cube roots.

Radical Expressions

Radical expressions include roots beyond just the square root; they can involve cube roots, fourth roots, or higher. A radical expression contains a radical symbol (\( \sqrt{} \)) and a radicand, which is the number under the radical.
Simplifying radical expressions involves reducing the expression to its simplest form while ensuring the radicand has no perfect power factors.
For cube roots, we often look for factors that are perfect cubes, which are numbers like 1, 8, 27, and so on. In cases where these factors are not present, the radical remains unchanged, just like in the expression \( \sqrt[3]{\frac{2}{3}} \).

• Simplification often involves factoring out perfect powers.
• The order of the root changes the approach to simplification (e.g., cube vs. square).
• Radicals can be rewritten using exponents: \( x^{\frac{1}{3}} \) is the same as \( \sqrt[3]{x} \).

Fraction Simplification

Simplifying fractions in radical expressions takes a slightly different approach than regular arithmetic fractions. With radicals, we often combine terms using rules for exponents and roots, like combining separate cube roots into a single radical expression.
When you have a fraction within a radical, such as \( \sqrt[3]{\frac{a}{b}} \), it means you're finding the cube root of the entire fraction.
To simplify such an expression:

• Find the cube roots of both the numerator and the denominator separately if possible.
• If they cannot be simplified further, as with our example \( \sqrt[3]{2} \) over \( \sqrt[3]{3} \), leave them in their respective radical form.
• Use the rule \( \frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}} \) to rewrite the expression.

These steps allow the simplification of complex expressions while maintaining the integrity of the radical expression. Ensure both components of the fraction are in their simplest form.