Problem 16 - College Algebra with Modeling and Visualization

Question

Find the cube root of the number. $$ 64 $$

Step-by-Step Solution

Answer

The cube root of 64 is 4.

Step 1: Understanding the Problem

We need to find the cube root of 64, which means identifying the number that, when multiplied by itself twice (i.e., three times), equals 64.

Step 2: Set Up the Equation

To solve for the cube root, we set up the equation: If \( x \) is the cube root of 64, then \( x^3 = 64 \).

Step 3: Solve for the Cube Root

Determine which number, when cubed, yields 64. If \( x^3 = 64 \), then by checking possible values, we find that \( x = 4 \) because \( 4^3 = 4 \times 4 \times 4 = 64 \).

Step 4: Check the Solution

Verify by cubing 4: Since \( 4^3 = 64 \), it confirms that the cube root of 64 is indeed 4.

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. These symbols typically represent numbers or variables, and algebra is used to express mathematical relationships. In the context of finding cube roots, algebra helps in setting up the equations that need solving. For instance, in the original exercise with the number 64, we use algebra to express the problem of finding the cube root as an equation:

• The number to find is represented as a variable, often denoted by \( x \).
• The problem to solve is \( x^3 = 64 \), where \( x^3 \) means \( x \) raised to the power of three, symbolizing a cube.

Using algebraic techniques, we then manipulate this equation to find the value of \( x \), resulting in the cube root we are looking for. With clear understanding and manipulation of algebraic symbols, the process becomes structured and efficient.

Equation Solving

Equation solving is a fundamental aspect of mathematics that involves finding unknown parameters that satisfy a given equation. For cube roots, the equation typically takes the form of \( x^3 = n \), where \( n \) is the number for which you want to find the cube root.

• To solve the equation \( x^3 = 64 \), you need to find a number \( x \) such that multiplying \( x \) by itself twice yields 64.
• This involves either algebraic manipulation or using trial methods to check various numbers until the correct one is found.

Once we have identified that \( x = 4 \) fits because \( 4 \times 4 \times 4 = 64 \), we solve the equation and find our cube root. The process requires logical reasoning and sometimes, if the number is less straightforward, the use of more advanced mathematical techniques like approximate methods or algorithms.

Exponents

Exponents are mathematical notations indicating the number of times a number, known as the base, is multiplied by itself. They are represented as a small number placed to the upper right of the base number. In our exercise, the exponent used is 3, indicating a cube, which is another term for a number raised to the power of three.

• The exercise of finding the cube root of 64 involved understanding and manipulating the expression \( 4^3 \).
• Here, the exponent 3 means the base number 4 is used in a multiplication sequence three times: \( 4 \times 4 \times 4 \).

Understanding exponents is crucial because they provide a concise way to represent repeated multiplication. Learning how to interpret and work with exponents allows us to tackle problems involving roots more effectively. Exponents simplify complex multiplication and, inversely, provide a structured method for finding roots when reversed.