Problem 16 - Fundamentals of Mathematics

Question

Determine the value of each power and root. \(\sqrt[3]{64}\)

Step-by-Step Solution

Answer

The cube root of 64 is 4.

Step 1: Analyze the Cube Root

The exercise asks us to find the cube root of 64, denoted by \( \sqrt[3]{64} \). Essentially, we need to determine what number, when multiplied by itself three times (cubed), equals 64.

Step 2: Identify Potential Cube Number

We identify which number, when cubed, is equal to 64. We can test smaller integer values to find \( n^3 = 64 \). Trying 4, we calculate \( 4 \times 4 \times 4 = 64 \). Since this equation holds, 4 is the correct value for the cube root of 64.

Step 3: Verify the Cube

We verify that our identified cube root is correct by multiplying 4 by itself three times to ensure the original base number is returned. Calculate: \( 4^3 = 4 \times 4 \times 4 = 64 \). Therefore, the calculation confirms that 4 is the right cube root.

Key Concepts

Power and Root

In mathematics, powers and roots are fundamental concepts that describe different ways of using numbers. When we talk about a "power," we are referring to the process called exponentiation, where a number (the base) is multiplied by itself a certain number of times. For instance, in the expression \( 4^3 \), 4 is the base, and it is raised to the power of 3, meaning it multiplies itself three times: \( 4 \times 4 \times 4 = 64 \).
In contrast, a "root" asks what base number, when multiplied by itself the indicated number of times, results in the given number. In this exercise, the cube root \( \sqrt[3]{64} \) asks: "What number, cubed, gives us 64?" The solution reveals that this number is 4 because \( 4^3 = 64 \). Understanding both powers and roots involves navigating inverses of operations—one building up (powers) and the other breaking down (roots). This provides a useful mathematical symmetry that can solve various problems.

Mathematical Operations

Mathematical operations are tools we use to manipulate numbers. These include addition, subtraction, multiplication, and division. In terms of roots, we introduce a more complex operation called finding the root itself. Specifically, cube roots involve finding which number multiplied by itself three times results in another number.
Take an example like cube rooting, which consists of solving equations like \( x^3 = 64 \). We do this by experimenting with numbers to determine what \( x \) satisfies the equation. When testing small positive integers, we find that \( 4 \times 4 \times 4 = 64 \), making 4 the cube root of 64.

• Cube roots ask for inverse operations of cubes.
• We're finding a base number from a result.
• These operations make it possible to reverse exponential growth.

It is selected based on feasibility to perform these operations mentally or with basic arithmetic, especially with lower numbers like 64.

Number Theory

Number theory is a branch of pure mathematics devoted to studying integers and integer-valued functions. It explores fascinating properties and relationships of numbers. In this context, cube roots like \( \sqrt[3]{64} \) are significant in analyzing the structure and behavior of numbers.
The concept of finding roots, particularly cube roots in number theory, involves recognizing how numbers can form perfect cubes. For 64, this translates to spotting that it can be factored as \( 4 \times 4 \times 4 \), making it a perfect cube of 4.

• The understanding of cube roots helps in factoring and simplifying equations.
• They can be crucial when solving integer equations in number theory.
• Cube roots are instrumental in understanding cubes' geometric and algebraic properties.

Deepening your understanding of numbers through such operations allows you to appreciate their intrinsic mathematical elegance and complexity.