Problem 52 - Algebra A Combined Function

Question

If \(x=-2, y=0\), and \(z=3\), find the value of each expression. $$ \frac{x^{3}-2 y+z}{2 z} $$

Step-by-Step Solution

Answer

The value of the expression is \(-\frac{5}{6}\).

Step 1: Substitute the Values

First, substitute the given values of \(x = -2\), \(y = 0\), and \(z = 3\) into the expression. The expression to evaluate is: \(\frac{x^3 - 2y + z}{2z}\). This becomes \(\frac{(-2)^3 - 2(0) + 3}{2(3)}\).

Step 2: Evaluate the Numerator

Now evaluate the numerator of the expression \((-2)^3 - 2(0) + 3\). Start with \((-2)^3\), which is \(-8\). Then, subtract \(2 \times 0\) (which is 0) from \(-8\), and add 3. This results in: \(-8 + 3 = -5\).

Step 3: Evaluate the Denominator

Evaluate the denominator, which is \(2z\). With \(z = 3\), this is \(2 \times 3 = 6\).

Step 4: Divide the Numerator by the Denominator

Finally, divide the result of the numerator by the denominator to find the value of the expression. This means dividing \(-5\) by \(6\), which gives \(-\frac{5}{6}\).

Key Concepts

Substitution

Substitution is the first step to simplifying expressions, and it involves replacing variables with their given values. This method makes it easier to evaluate expressions since you're dealing with known numbers instead of variables. For example, in the expression \(\frac{x^3 - 2y + z}{2z}\), if you're given \(x = -2\), \(y = 0\), and \(z = 3\), you substitute these values into the expression.

• Replace \(x\) with -2.
• Replace \(y\) with 0.
• Replace \(z\) with 3.

Substituting leads to \(\frac{(-2)^3 - 2(0) + 3}{2(3)}\), which simplifies the task by turning it into mere arithmetic operations rather than working with unknowns.

Numerators and Denominators

In fractions, understanding numerators and denominators is crucial as they form the backbone of fraction operations. The numerator is the top part of a fraction, while the denominator is the bottom part. They represent the division operation intrinsic to fractions.To evaluate our given expression:

• The numerator is \((-2)^3 - 2(0) + 3\).
• The denominator is \(2z\), or more specifically \(2 \times 3\) once substitution is applied.

By performing operations separately on the numerator and denominator, you understand how each affects the overall value of the fraction.

Exponentiation

Exponentiation is a mathematical operation involving a base and an exponent. It tells us how many times to multiply the base by itself. In our expression, \((-2)^3\) represents exponentiation with -2 as the base and 3 as the exponent.

• The exponent 3 means \(-2\) needs to be multiplied by itself three times: \((-2) \times (-2) \times (-2)\).
• This results in -8, because multiplying three negatives results in a negative product.

Understanding this step is important, as exponentiation drastically changes values, especially when dealing with negative bases!

Division of Fractions

The division of fractions involves determining how many times the denominator fits into the numerator. Once the numerator and denominator have been evaluated, division can simplify the expression to a single number.

• For instance, with a numerator of -5 and a denominator of 6, you're essentially finding the quotient of \(-5 \div 6\).
• This results in \(-\frac{5}{6}\), which is your simplified expression.

Remember, the sign of the resulting fraction is determined by the signs of the numerator and the denominator. If one is negative and the other is positive, the result is negative. This step provides the final evaluated value of the expression.