Problem 98 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt[3]{\frac{4}{125}} $$

Step-by-Step Solution

Answer

\( \sqrt[3]{\frac{4}{125}} = \frac{\sqrt[3]{4}}{5} \)

Step 1: Understand the Expression

We need to simplify the expression \( \sqrt[3]{\frac{4}{125}} \). This is the cube root of the fraction \( \frac{4}{125} \). The goal is to express this in simplest form.

Step 2: Simplify the Denominator

Notice that 125 is a perfect cube because \( 125 = 5^3 \). Thus, the cube root of 125 is 5. So, \( \sqrt[3]{125} = 5 \).

Step 3: Simplify the Numerator

The numerator is 4, which is not a perfect cube. Therefore, the cube root of 4 will remain \( \sqrt[3]{4} \).

Step 4: Combine Results

Combine the simplified results from Steps 2 and 3. The expression \( \sqrt[3]{\frac{4}{125}} \) simplifies to \( \frac{\sqrt[3]{4}}{5} \).

Key Concepts

Cube Root Simplification

The cube root of a number is a special root that asks, "What number, when multiplied by itself three times, gives me this number?" It is slightly more complex than the square root.
In this exercise, we are working with the cube root of a fraction, \( \sqrt[3]{\frac{4}{125}} \). The cube root can be applied separately to the numerator and the denominator because of the property of radicals: \( \sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}} \).

• Step 1: Identify if the denominator or numerator is a perfect cube. A perfect cube is a number you can get by raising another number to the third power.
• Step 2: Simplify the cube root of any perfect cubes.
• Step 3: Leave the cube root of non-perfect cubes as-is, if they cannot be simplified further.

In the given exercise \( 125 \) was identified as a perfect cube, as \( 5^3 = 125 \). Meanwhile, 4 was not a perfect cube, so \( \sqrt[3]{4} \) remains in its radical form.

Fractional Radicals

Fractional radicals involve taking the root of a fraction, like \( \sqrt[3]{\frac{4}{125}} \). The main strategy is to treat the numerator and denominator separately.
By simplifying each part individually, we get a clearer view of how the expression behaves.

• Focus on the Denominator: For instance, with the denominator \( 125 \), we know this is \( 5^3 \). Thus, \( \sqrt[3]{125} = 5 \).
• Focus on the Numerator: In our case, \( 4 \) does not break down into a perfect cube nicely, so it stays as \( \sqrt[3]{4} \).

By treating each part of the fraction separately, you can effectively simplify the overall radical expression. The resulting form is \( \frac{\sqrt[3]{4}}{5} \), showing a tidy division of interests between the numerator and denominator.

Algebraic Expressions

Understanding algebraic expressions is essential when simplifying exercises like these. They combine numbers, roots, and variables we manipulate to find simpler forms.
Here, we simplified the algebraic expression \( \sqrt[3]{\frac{4}{125}} \).
Steps in Tidying Expressions:

• Factor and Reduce: For radical expressions, factoring numbers helps us see simpler cube roots and reduces fractions where possible.
• Keep a Clean Format: Write expressions clearly, avoiding clutter to maintain mathematical clarity. Utilize radical simplifications and fractions efficiently.

Applying these concepts properly allows us to see that \( \frac{\sqrt[3]{4}}{5} \) is the simplest form of \( \sqrt[3]{\frac{4}{125}} \). It’s a cleaner result, giving a more concise expression that's easier to understand when re-encountered.