Problem 72 - College Algebra

Question

Explain the best way to evaluate \(\frac{900 !}{899 !}\) without calculator.

Step-by-Step Solution

Answer

The best way to evaluate \(\frac{900 !}{899 !}\) without a calculator is, simplifying it by cancelling out 899!, leaving you with 900.

Step 1: Identify the Key Factorial Relationship

We know from theory that factorials have a particular relationship: n! = n*(n-1)!. This means we could express 900! as 900*899!.

Step 2: Apply the Factorial Relationship to the Numerator

Replace 900! in the numerator by 900*899!. So, the fraction becomes \(\frac{900 * 899!}{899!}\)

Step 3: Cancel Out Similar Terms

899! is present in both the numerator and the denominator. Cancel them out to simplify the fraction. We now have just 900 left.

Key Concepts

Factorial Simplification

Understanding the concept of factorial simplification is essential when dealing with expressions like \( \frac{900!}{899!} \). Factorials, denoted by an exclamation mark, are products of all positive integers up to a given number. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).

In problems involving factorials, it's often necessary to simplify expressions to make calculations manageable without a calculator. One key property to remember is:

• \( n! = n \times (n-1)! \)

This tells us that for any positive integer \( n \), the factorial can be broken down into simpler components.

In our original problem, this simplification process is crucial. By expressing \( 900! \) as \( 900 \times 899! \), we can then cancel out \( 899! \) from the numerator and the denominator, drastically simplifying our work, down to the number 900. This is the essence of factorial simplification.

Algebraic Manipulation

Algebraic manipulation is all about rearranging and simplifying mathematical expressions to find the value or form you desire. When working with factorials, this often involves recognizing patterns and using them to your advantage.

In the problem \( \frac{900!}{899!} \), the first step of manipulating the expression involves recognizing the relationship of factorials and rewriting \( 900! \) as \( 900 \times 899! \). This understanding allows us to rewrite complicated expressions in a form that allows terms to cancel or simplify.

Next, once rewritten, the \( 899! \) terms in the numerator and the denominator can be cancelled. This method employs the fundamental algebraic principle that allows an equation to be simplified by removing like terms.

By performing these steps, you've effectively reduced a seemingly complex expression into a very simple one, using the power of algebraic manipulation. This skill is key not just in factorial problems, but in all areas of mathematics where simplifying expressions can lead to solutions more easily understood and calculated.

Number Theory

Number theory is the study of properties and relationships of numbers, particularly focusing on integers and whole numbers. Factorials often show up in number theory because they involve the multiplication of sequences of whole numbers.

In the context of our problem, understanding how factorials work in number theory helps to simplify expressions and recognize patterns. The simplification of \( \frac{900!}{899!} \) is a beautiful example of how number properties can reduce complexity.

Within number theory, concepts such as divisibility and factorial properties allow us to recognize why and how terms cancel in an expression. In our scenario, knowing that \( 900! \) can be expressed as \( 900 \times 899! \) uses these principles. The factorial simplification shows a direct application of number theory to remove unnecessary calculations.

As you progress in number theory, these foundational skills—scrutinizing numbers for patterns and simplifying using known properties—will reveal more elegance and order in mathematical relationships and problems. Understanding these core concepts can open up a deeper appreciation and broader applications in mathematics and beyond.