Problem 31 - Algebra 1

Question

Simplify the expression. $$\sqrt[3]{\frac{3}{16}}$$

Step-by-Step Solution

Answer

The simplified expression of \(\sqrt[3]{\frac{3}{16}}\) is approximately \(\frac{\sqrt[3]{3}}{2.52}\).

Step 1: Identify the cube root of the numerator and denominator

Identify the cube root of the numerator, which is 3, and the cube root of the denominator, which is 16. The cube root of a number a is the number b, such that \(b^3 = a\).

Step 2: Calculate the cube roots

Calculate the cube roots: The cube root of 3 cannot be simplified further, so it stays as it is. The cube root of 16 is not a round number, but it can be presented as \(2.52\), rounded to two decimal places.

Step 3: Write down the simplified cube root expression

Write down the simplified fraction. The cube root of \(\frac{3}{16}\) can be presented as \(\frac{\sqrt[3]{3}}{2.52}\).

Key Concepts

Understanding Numerator and Denominator

In any fraction, we have two main parts: the numerator and the denominator. The numerator is the top number and represents how many parts of a whole are being considered. The denominator is the bottom number and tells us into how many equal parts the whole is divided. For a fraction like \( \frac{3}{16} \), 3 is the numerator and 16 is the denominator. Understanding these components is crucial, especially in operations like finding cube roots, because you need to consider both parts separately. Each part has its own independent significance in determining the overall value of the fraction.

Fundamentals of Calculating Cube Roots

Calculating cube roots involves finding a number that, when multiplied by itself twice, gives the original number. For example, for a number \( a \), the cube root \( \sqrt[3]{a} \) is a number \( b \) such that \( b^3 = a \).When calculating the cube roots for fractions, you handle the numerator and denominator individually:

• Find the cube root of the numerator.
• Find the cube root of the denominator.

The cube root of 3, in our exercise, does not have a simple whole number answer. We keep it as \( \sqrt[3]{3} \). For the denominator 16, the cube root is approximately 2.52, highlighting the need for decimal approximation when a neat root doesn't exist.

Expressing Cube Roots in Decimal Form

In many cases, cube roots do not result in neat whole numbers. In such instances, expressing cube roots in decimal form becomes valuable. It allows for a more practical and readable format, particularly when precise equations are required in real-world applications.In our example, the cube root of the fraction \( \frac{3}{16} \) led us to an expression \( \frac{\sqrt[3]{3}}{2.52} \). Here, 2.52 is a decimal approximation of the cube root of 16, rounded to two decimal places.Using decimal form entails:

• Improving readability and understanding, especially for complex numbers.
• Simplifying use in calculations.
• Ensuring a level of accuracy that's sufficient for many practical contexts.

Calculators and specialized software are often used to find these decimal values for cube roots, ensuring precision.