Problem 3 - Algebra A Combined Function

Question

Find the value of the following expressions when \(x=2, y=-2,\) and \(z=-5\). $$ \frac{y^{3}}{y^{2}-1} $$

Step-by-Step Solution

Answer

The value of the expression is \(-\frac{8}{3}\).

Step 1: Substitute Values

First, substitute the given values of \(y\) and \(z\) into the expression. Since \(x\) is not in the expression, we only substitute \(y=-2\) into \(\frac{y^{3}}{y^{2}-1}\).

Step 2: Calculate the Numerator

The numerator of the expression is \(y^3\). Substitute \(y = -2\) to get \((-2)^3 = -8\).

Step 3: Calculate the Denominator

The denominator of the expression is \(y^2 - 1\). Substitute \(y = -2\) to calculate \((-2)^2 - 1 = 4 - 1 = 3\).

Step 4: Solve the Expression

Now that we have the values for the numerator and the denominator, substitute them into the expression: \(\frac{-8}{3}\).

Step 5: Simplify the Fraction

The fraction \(\frac{-8}{3}\) is already in its simplest form as the numerator and the denominator do not have any common factors other than 1.

Key Concepts

Substitution Method

The substitution method is an essential technique in algebra, especially when solving expressions with variables. To use this method, we replace each variable with its given value. In the problem at hand, we need to find the value of \[ \frac{y^{3}}{y^{2}-1} \] when \(y = -2\). Here, the variable \(y\) is given a specific integer value. By substituting \(-2\) for \(y\), you craft a numeric equation instead of one with multiple abstract variables. This makes operations like addition, subtraction, multiplication, and division easier to perform as they are now numeric rather than algebraic.

Exponents

Exponents are a shorthand way of expressing repeated multiplication. For example, \(y^3\) means \(y\) is multiplied by itself three times. In this exercise, after substituting \(y = -2\), we calculate:

• \((-2)^3\): This means multiplying \(-2\) by itself three times, equalling \(-2 \times -2 \times -2 = -8\).
• \((-2)^2\): This involves multiplying \(-2\) by itself two times, which calculates as \(-2 \times -2 = 4\).

Recognizing how exponents work with negative numbers is crucial. Odd exponents retain the negative sign while even exponents do not. This is because multiplying two negative numbers results in a positive number, but if a third is multiplied, the result shifts to negative again.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. In this problem, we obtained the fraction \[ \frac{-8}{3} \] after substituting and evaluating both the numerator \((-8)\) and the denominator \(3\). A fraction is simplified when the numerator and the denominator have no common factors other than 1. Here:

• The greatest common divisor of \(-8\) and \(3\) is 1.
• Since the numbers are relatively prime, \(\frac{-8}{3}\) is the most simplified version of the fraction.

Always attempt to simplify your fractions unless asked otherwise, making them cleaner and easier to interpret in further mathematical operations.