Problem 26 - Fundamentals of Mathematics

Question

Is 3 a factor of \(2^{6} \cdot 3^{2} \cdot 5^{3} \cdot 4^{6} ?\) Explain.

Step-by-Step Solution

Answer

Yes, 3 is a factor.

Step 1: Simplify the Expression

First, simplify the expression given: \[2^{6} \cdot 3^{2} \cdot 5^{3} \cdot 4^{6}\]Notice that \(4\) can be rewritten in terms of \(2\) as \(2^{2}\). So, \(4^{6}\) becomes \((2^{2})^{6}\), which simplifies to \(2^{12}\). Thus, the expression becomes:\[2^{6} \cdot 3^{2} \cdot 5^{3} \cdot 2^{12}\]

Step 2: Combine Like Terms

Combine the powers of 2 in the expression:\[2^{6}\cdot 2^{12} = 2^{6+12} = 2^{18}\]The fully simplified expression is then:\[2^{18} \cdot 3^{2} \cdot 5^{3}\]

Step 3: Check for Factor of 3

Since we are checking if 3 is a factor of the expression, look for the power of 3 in this expression:\[2^{18} \cdot 3^{2} \cdot 5^{3}\]Here, the power of 3 is 2, which means 3 is indeed a factor of this expression as it appears in the product.

Key Concepts

Factors

In mathematics, a factor is a number that divides another number without leaving a remainder. When we say that "3 is a factor," we are essentially stating that 3 can be multiplied by another whole number to yield our target number.
For instance, in the expression we analyzed, \[2^{18} \cdot 3^{2} \cdot 5^{3}\]We were tasked with checking if 3 is a factor. To determine this, we examined where the number 3 appeared in the factorization. We found that 3 was raised to the power of 2, meaning within the terms of multiplication, 3 is indeed a factor.

Another way to think about factors is by visualizing them as the "building blocks" of a number. Every number can be broken down into its basic components or factors. Understanding factors is crucial as it helps in simplifying expressions and solving equations. It lays the foundation for concepts like Greatest Common Divisor (GCD) and Least Common Multiple (LCM), which are important in arithmetic and algebra.

Exponents

Exponents represent repeated multiplication of a number by itself. In simple terms, the exponent tells you how many times to multiply a number, known as the base, by itself. For example, in the expression \[3^{2}\], "3" is the base, and "2" is the exponent, indicating that 3 is multiplied by itself once:\[3 \times 3 = 9\].Exponents allow for compact representation of large numbers or complex multiplications.

In our exercise, we handled multiple terms that involved exponents, namely:

• \(2^{18}\)
• \(3^{2}\)
• \(5^{3}\)

When simplifying expressions with exponents, one can add the exponents if the base is the same, as seen with our factors of 2. For instance:\[2^{6} \times 2^{12} = 2^{18}\],showing how these powerful notations can greatly ease mathematical operations and aid in solving complex problems efficiently.

Mathematical Expressions

A mathematical expression is a combination of numbers, variables, and operators, such as addition and multiplication, arranged in a meaningful way. Expressions are often used for computations, simplifications, and deriving mathematical relations.The given expression in the problem,\[2^{6} \cdot 3^{2} \cdot 5^{3} \cdot 4^{6}\],illustrates how expressions can be composed of different elements. Here it includes numbers and operations involving exponents.

The exercise required us to simplify this expression by recognizing and restructuring it, where needed. By understanding that 4 can be translated to \[(2^{2})\],we could further simplify \[4^{6}\] into \[(2^{2})^{6} = 2^{12}\]. This kind of manipulation demonstrates the flexibility and power of expressions in conveying complex operations clearly.Mathematical expressions are an essential tool for solving equations, optimizing problems, and describing geometric shapes. They are the language of math, a concise way to communicate complex ideas succinctly and accurately.