Problem 46 - Algebra and Trigonometry

Question

\(39-48=\) Simplify the expression. $$ \sqrt[3]{24}-\sqrt[3]{81} $$

Step-by-Step Solution

Answer

The expression \( \sqrt[3]{24} - \sqrt[3]{81} \) simplifies to \(-1.443\) if evaluated numerically.

Step 1: Understand the Cube Root

The cube root of a number \( x \), denoted \( \sqrt[3]{x} \), is the number \( y \) such that \( y^3 = x \). This means we need to find a number which, when multiplied by itself three times, gives \( x \).

Step 2: Approximate Cube Roots

The cube root of 24, \( \sqrt[3]{24} \), is between 2 and 3 because \( 2^3 = 8 \) and \( 3^3 = 27 \). Similarly, the cube root of 81, \( \sqrt[3]{81} \), is between 4 and 5 because \( 4^3 = 64 \) and \( 5^3 = 125 \).

Step 3: Simplify the Expression

The expression \( \sqrt[3]{24} - \sqrt[3]{81} \) cannot be simplified to a single number without estimation because neither 24 nor 81 are perfect cubes. Therefore, the expression represents the difference between the two approximated cube roots computed.

Step 4: Numerical Approximation (Optional)

For practical purposes, using a calculator: \( \sqrt[3]{24} \approx 2.884 \) and \( \sqrt[3]{81} \approx 4.327 \). Thus, the numerical evaluation is approximately \( 2.884 - 4.327 \approx -1.443 \).

Key Concepts

Perfect Cubes

A perfect cube is a number that can be expressed as the product of an integer multiplied by itself twice. For example, numbers like 8, 27, and 64 are termed as perfect cubes because they can be written as the cube of 2, 3, and 4 respectively:

• 8 is a perfect cube because \( 2 \times 2 \times 2 = 8 \).
• 27 is a perfect cube because \( 3 \times 3 \times 3 = 27 \).
• 64 is a perfect cube as \( 4 \times 4 \times 4 = 64 \).

In contrast, numbers like 24 and 81 are not perfect cubes. They cannot be expressed neatly as the cube of an integer. This lack of easy factorization is why their cube roots are not simple integers. Recognizing whether a number is a perfect cube or not is crucial, as it affects the accuracy and method of calculating various algebraic expressions.

Approximation of Cube Roots

Finding the cube root of a non-perfect cube like 24 or 81 requires approximation methods. We start by identifying the perfect cubes nearest to the number in question. For instance:

• The cube root of 24, \( \sqrt[3]{24} \), lies between the perfect cubes 8 (\(2^3\)) and 27 (\(3^3\)).
• The cube root of 81, \( \sqrt[3]{81} \), falls between 64 (\(4^3\)) and 125 (\(5^3\)).

By narrowing the range, we can more accurately approximate the cube root using a calculator or manual computation methods. While a perfect calculation may not be feasible without precise tools, knowing the range allows for a reasonable estimate and better understanding of the problem.

Simplifying Expressions

Simplifying expressions involving cube roots focuses on reducing the expression to its most understandable form, often by using approximation if necessary. In the problem, the expression \( \sqrt[3]{24} - \sqrt[3]{81} \) doesn't simplify into a neat integer because neither operand is a perfect cube. The expression essentially demonstrates the difference between two estimated cube roots.

• We acknowledge the intricate nature of the numbers involved.
• Using approximation, we calculate that \( \sqrt[3]{24} \approx 2.884 \) and \( \sqrt[3]{81} \approx 4.327 \).
• The expression simplifies to approximately \( 2.884 - 4.327 \approx -1.443 \).

It's important to know that expressions may not always resolve into simple numbers, emphasizing the importance of understanding approximation and accepting an estimated result.