Problem 87 - Elementary Algebra

Question

Find the value of each of the following expressions. \(P=\frac{n(n-2)(n-4)}{2 n} . \quad\) Find \(P\) if \(n=-6\)

Step-by-Step Solution

Answer

Answer: -40

Step 1: Substitute the value of n in the expression

Substitute n = -6 in the expression \(P=\frac{n(n-2)(n-4)}{2 n}\). \(P=\frac{-6(-6-2)(-6-4)}{2 (-6)}\)

Step 2: Simplify the expression

Simplify the expression by calculating the values inside the parentheses. \(P=\frac{-6(-8)(-10)}{2 (-6)}\)

Step 3: Multiply the terms in the numerator and denominator

Calculate the numerator and denominator by multiplying the respective terms together. \(P=\frac{480}{-12}\)

Step 4: Divide the numerator by the denominator

Divide 480 by -12 to find the final value of P. \(P=-40\) Therefore, the value of \(P\) when \(n=-6\) is \(-40\).

Key Concepts

Substitution

Substitution is an important concept in algebra that involves replacing variables with numbers or other expressions. In our exercise, we substitute the given value of \(n = -6\) into the algebraic expression for \(P\): \(P=\frac{n(n-2)(n-4)}{2n}\).
Here’s how to do it step-by-step:

• First, identify the variable in the expression, which is \(n\).
• Next, replace every instance of \(n\) in the expression with the number \(-6\).
• Your expression now looks like this: \(P=\frac{-6(-6-2)(-6-4)}{2(-6)}\).

This substitution bridges the gap between theoretical math and calculable numbers. Substitution allows us to transition from an algebraic expression into something more concrete that we can compute.

Simplifying Expressions

Simplifying expressions is the process of transforming an expression into its simplest form. It often involves reducing the complexity while keeping the expression equivalent. Let's see how it’s done in our example.Firstly, once we substitute \(n = -6\), our expression becomes \(P=\frac{-6(-8)(-10)}{2 (-6)}\).
The task now is to simplify it:

• Begin by performing operations inside the parentheses: \(-6-2 = -8\), and \(-6-4 = -10\).
• The expression then transforms to multiplied terms: \(-6\), \(-8\), and \(-10\).

In essence, simplifying expressions involves performing arithmetic operations and restructuring an expression into its most manageable form. This makes the calculation straightforward and aids in understanding.

Numerator and Denominator Manipulation

Manipulating the numerator and denominator is a crucial skill when working with fractions in algebra. This process involves performing operations to simplify the fraction's top (numerator) and bottom (denominator) parts separately, before combining them.In our original exercise:

• The numerator is \(-6 \times -8 \times -10\).
• The denominator is \(2 \times -6\).
• First, calculate the product of the numbers in the numerator: \(-6 \times -8 \times -10 = 480\).
• Next, calculate the product in the denominator: \(2 \times -6 = -12\).

After computing both, the expression becomes \(\frac{480}{-12}\). Finally, you perform the division to simplify the entire fraction and arrive at the solution \(P = -40\). Understanding how to manipulate numerators and denominators is essential for simplifying fractions and arriving at the correct answer.