Problem 2 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$-\sqrt[3]{64}$$

Step-by-Step Solution

Answer

The expression evaluates to \(-4\).

Step 1: Understand Cube Roots

The expression involves finding the cube root, written as \( \sqrt[3]{64} \). Cube roots are numbers that, when multiplied by themselves twice (a total of three times including the original number), give the original number. For example, if \( x = \sqrt[3]{y} \), then \( x^3 = y \).

Step 2: Identify Perfect Cubes

To solve \( \sqrt[3]{64} \), we need to determine which number, when cubed, equals 64. Recognize that\( 4 \times 4 \times 4 = 4^3 = 64 \). Thus, \( \sqrt[3]{64} = 4 \).

Step 3: Apply the Negative Sign

The original expression \( -\sqrt[3]{64} \) includes a negative sign outside the cube root. After finding \( \sqrt[3]{64} = 4 \), apply the negative sign to result in \( -4 \).

Step 4: Conclude the Evaluation

Combine the results from the previous steps to conclude that \( -\sqrt[3]{64} = -4 \). This expression evaluates to \(-4 \).

Key Concepts

Cube Roots

Understanding cube roots can demystify many algebraic problems. Cube roots are essentially the opposite of cubing a number. When you find the cube root of a number, you are looking for a value that, when multiplied by itself three times, returns to the original number. For instance, the cube root of 27 is 3, because multiplying 3 by itself twice yields 27: \( 3 \times 3 \times 3 = 27 \).

• The cube root symbol is written as \( \sqrt[3]{} \), located to the left of the number you wish to cube root.
• Finding the cube root of perfect cubes is straightforward since they yield whole numbers.
• In contrast, cube roots of non-perfect cubes result in more complex, often irrational numbers.

To find the cube root, aim to decompose the number into smaller, identical factors three times.

Perfect Cubes

Identifying perfect cubes is crucial when working with cube roots. A perfect cube is a number that is the result of taking an integer and multiplying it by itself three times. Recognizing perfect cubes helps simplify cube root calculations, where you can quickly determine the root without extra steps.
Common perfect cubes include:

• \(1^3 = 1\)
• \(2^3 = 8\)
• \(3^3 = 27\)
• \(4^3 = 64\)
• \(5^3 = 125\)

Each of these is a product of an integer multiplied by itself twice more. Knowing these values by heart can greatly speed up calculations involving cube roots, as you can immediately identify the integer that produces these perfect cubes.

Mathematical Expressions

Mathematical expressions involve combinations of numbers, variables, and operations like addition, multiplication, or in this case, roots. The expression \( -\sqrt[3]{64} \) showcases how cube roots function within a broader mathematical context.
When handling mathematical expressions, always note:

• Order of operations, which dictates how calculations proceed. With cube roots, evaluate the root first before applying any external operations like addition or multiplication.
• The impact of signs. In the expression \( -\sqrt[3]{64} \), the negative sign indicates inversion after the cube root extraction.
• Understanding the properties of numbers used within the expression, especially when working with roots, can simplify the process.

In tackling exercises involving cube roots, decomposing the expression into understandable steps helps solidify your grasp of the underlying math principles.