Problem 20 - Algebra 2

Question

Simplify. $$ \sqrt[3]{\frac{1}{125}} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt[3]{\frac{1}{125}} \) is \( \frac{1}{5} \).

Step 1: Understand the Problem

You need to simplify the expression \( \sqrt[3]{\frac{1}{125}} \). This means you are looking for the cube root of \( \frac{1}{125} \). The cube root of a number \( x \) is another number \( y \) such that \( y^3 = x \).

Step 2: Simplify the Denominator

Identify the cube root of a perfect cube in the denominator. Here, \( 125 \) is a perfect cube because \( 5^3 = 125 \). Thus, \( \sqrt[3]{125} = 5 \).

Step 3: Calculate the Cube Root

Now apply the cube root to the numerator and denominator separately. Since \( \sqrt[3]{1} = 1 \) and we already know \( \sqrt[3]{125} = 5 \), the expression becomes \( \frac{\sqrt[3]{1}}{\sqrt[3]{125}} = \frac{1}{5} \).

Step 4: Verify the Simplification

Check if \( \left( \frac{1}{5} \right)^3 \) equals \( \frac{1}{125} \). Indeed, \( \left( \frac{1}{5} \right)^3 = \frac{1^3}{5^3} = \frac{1}{125} \), confirming that the simplification is correct.

Key Concepts

Cube Root

A cube root is an operation used to determine a number that produces a given number when raised to the power of three. It is symbolically represented as \( \sqrt[3]{x} \). For example, if you have \( y \), and when you cube it (\( y^3 \)), it results in \( x \), then \( y \) is the cube root of \( x \).

When simplifying radical expressions where cube roots are involved, you look for the number which, when multiplied by itself twice (i.e., cubed), results in the original number under the radical sign. For instance, when tasked with finding \( \sqrt[3]{8} \), the answer would be \( 2 \), because \( 2 \times 2 \times 2 = 8 \).

• Cube roots can be used to simplify expressions involving irrational numbers.
• They are particularly useful in reducing expressions to simpler fractional values.

In our exercise, the cube root operation was applied to both the numerator and the denominator separately, leading to a simplified expression.

Perfect Cube

A perfect cube is a number that can be expressed as the cube of an integer. For example, numbers like 1, 8, 27, and 64 are perfect cubes because they can be written as \( 1^3 \), \( 2^3 \), \( 3^3 \), and \( 4^3 \) respectively.

Recognizing perfect cubes is essential in simplifying radical expressions, especially when working with cube roots. If your expression includes a perfect cube, it can be simplified easily as it becomes evident what the cube root is.

• In our given problem, the number 125 is identified as a perfect cube because \( 5^3 = 125 \).
• Knowing this helps us directly calculate the cube root: \( \sqrt[3]{125} = 5 \).

Finding a perfect cube often simplifies the task of reducing radical expressions, making the solutions more straightforward and understandable.

Numerator and Denominator

In a fraction, the numerator represents the top number, while the denominator represents the bottom number. These components must be understood when simplifying any radical expression that is in fractional form.

When dealing with cube roots in a fraction, you apply the cube root to both the numerator and the denominator separately. This is because taking the cube root of a fraction necessitates extracting the cube root individually from both components.

• For instance, to simplify \( \sqrt[3]{\frac{1}{125}} \), we find \( \sqrt[3]{1} \) and \( \sqrt[3]{125} \) separately.
• By doing this, the fraction simplifies to \( \frac{1}{5} \) because \( \sqrt[3]{1} = 1 \) and \( \sqrt[3]{125} = 5 \).

This approach of treating the numerator and denominator individually ensures clarity in simplification and helps avoid errors in calculations. It leads to a tidy and comprehensible final result.