Problem 40 - Algebra 2

Question

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[3]{\frac{8}{216}} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt[3]{\frac{8}{216}} \) is \( \frac{1}{3} \).

Step 1: Identifying Perfect Cubes

In this expression, \( \sqrt[3]{\frac{8}{216}} \), you should identify 8 and 216 as perfect cube numbers. 8 is \( 2^3 \) and 216 is \( 6^3 \).

Step 2: Apply Cube Root

After identifying perfect cubes, apply the cube root operation to both the numerator and the denominator separately. So, \( \sqrt[3]{\frac{8}{216}} \) becomes \( \frac{\sqrt[3]{8}}{\sqrt[3]{216}} \), which in turn simplifies to \( \frac{2}{6} \).

Step 3: Simplify Fraction

Finally, simplify the fraction \( \frac{2}{6} \) to its lowest terms, which is \( \frac{1}{3} \).

Key Concepts

Perfect Cubes

When dealing with radical expressions, understanding perfect cubes is essential. A perfect cube is a number that can be expressed as the cube of an integer. For example, 8 is a perfect cube because it can be written as \( 2^3 \). Similarly, 216 is another perfect cube because it is \( 6^3 \).

Recognizing perfect cubes allows us to simplify the expression easily when performing operations involving cube roots.

• To identify perfect cubes, think of small numbers raised to the power of 3.
• Common perfect cubes include \( 1^3 = 1 \), \( 2^3 = 8 \), \( 3^3 = 27 \), and so on.
• If the number is large, try to factor it out to smaller units and check if it is a cube of another number.

Identifying perfect cubes in both the numerator and the denominator is a crucial initial step before performing cube root operations.

Cube Root

Once you identify the numbers as perfect cubes, the next step is to apply the cube root operation. A cube root essentially "undoes" the effect of cubing a number. Hence, taking the cube root of a perfect cube gives back the original number.

In our example, we take the cube root separately:

• The cube root of 8 (\( \sqrt[3]{8} \) ) results in 2 because \( 2^3 = 8 \).
• Similarly, the cube root of 216 (\( \sqrt[3]{216} \) ) gives us 6 because \( 6^3 = 216 \).

Thus, the expression is simplified from \( \sqrt[3]{\frac{8}{216}} \) to \( \frac{\sqrt[3]{8}}{\sqrt[3]{216}} \), yielding \( \frac{2}{6} \).

Understanding cube roots simplifies solving problems involving radical expressions by reducing numbers into their most manageable form.

Fraction Simplification

After applying cube roots, simplifying fractions helps in attaining the final, simplest form of the expression. A simplified fraction has no common factors between the numerator and the denominator other than 1.

In this exercise, once you find that the fraction is \( \frac{2}{6} \), the next step is to simplify:

• To do this, find the greatest common divisor (GCD) of 2 and 6.
• Since both are divisible by 2, divide both the numerator and denominator by 2.
• This results in \( \frac{1}{3} \), which is the simplest form, as there are no common factors left.

Simplification is essential to present the expression in its lowest terms, ensuring clarity and precision in mathematical expressions. With practice, simplifying fractions becomes a quick and intuitive process.