Problem 28 - Precalculus Mathematics for Calculus

Question

Evaluate the expression using \(x=3, y=4,\) and \(z=-1\). $$(x y)^{2 z}$$

Step-by-Step Solution

Answer

The evaluated expression is \(\frac{1}{144}\).

Step 1: Substitute the Given Values

Substitute the values of \(x\), \(y\), and \(z\) into the expression. The expression is \((xy)^{2z}\). Using the given values \(x = 3\), \(y = 4\), and \(z = -1\), we substitute these into the expression:\((3 \times 4)^{2 \times (-1)}\).

Step 2: Simplify Inside the Parentheses

First, multiply the values inside the parentheses: \(3 \times 4 = 12\). So the expression becomes \(12^{2 \times (-1)}\).

Step 3: Calculate the Exponent

Multiply the numbers in the exponent: \(2 \times (-1) = -2\). Therefore, the expression now is \(12^{-2}\).

Step 4: Evaluate the Exponent

Evaluate \(12^{-2}\) by calculating the reciprocal: \(12^{-2} = \frac{1}{12^2}\). Find \(12^2\) which equals \(144\). So, \(12^{-2} = \frac{1}{144}\).

Key Concepts

Substitution

Substitution is a fundamental step in algebra, where you replace variables with their given numerical values. In the exercise, we have the expression \((xy)^{2z}\) and the variables are assigned as \(x = 3\), \(y = 4\), and \(z = -1\). By substituting, the entire expression becomes numerical, making it easier to evaluate. Here’s how it works:

• Identify the variables in the expression.
• Replace each variable with its assigned number.

By transforming the original expression into \((3 \times 4)^{2 \times (-1)}\), the substitution process helps in setting the stage for evaluating the expression further. Each step depends on correctly substituting these values, which simplifies the equation in manageable pieces.

Negative Exponents

Understanding negative exponents is crucial, as they often appear in algebraic expressions. A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent.For instance, if you have an expression like \(a^{-n}\), it can be rewritten as \(\frac{1}{a^n}\). Applying this to our exercise after substitutions gives us \(12^{-2}\), which translates into \(\frac{1}{12^2}\).Key points about negative exponents:

• They suggest inversion; the base is flipped into the denominator.
• Calculate the positive exponent first before finding the reciprocal.

In the solution, by evaluating the exponent \(-2\), we essentially compute \(12\) squared and then take the reciprocal, simplifying to \(\frac{1}{144}\). This method reveals how negative exponents transform and simplify expressions.

Evaluation of Expressions

Evaluating expressions means finding their numerical value, given specific values for the variables. It involves several steps, where substitution is immediately followed by managing arithmetic within the expression.In this exercise, after successfully substituting \(x = 3\), \(y = 4\), and \(z = -1\) into \((xy)^{2z}\), we next handle the arithmetic operations to simplify the expression. Consider this process:

1. Perform any multiplication inside the parentheses: \(3 \times 4 = 12\).
2. Compute the exponent process by first multiplying: \(2 \times (-1) = -2\).
3. Finally, turn \(12^{-2}\) into its reciprocal form: \(\frac{1}{144}\)

This methodical sequence translates the algebraic expression into a precise numerical value, following rules of exponents and arithmetic simplification, ensuring the correct evaluation of the expression.