Problem 22 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\(\frac{1}{144}\)

Step 1: Substitute Values

First, substitute the given values into the expression. The expression is \((xy)^{2z}\). Replace \(x\) with 3, \(y\) with 4, and \(z\) with -1. The expression becomes \((3 \times 4)^{2(-1)}\).

Step 2: Simplify Inside the Parentheses

Multiply the numbers inside the parentheses. Evaluate \(3 \times 4\), which equals 12. Now, the expression is \(12^{2(-1)}\).

Step 3: Simplify the Exponent

Evaluate the exponent \(2(-1)\) which equals -2. Thus, the expression simplifies further to \(12^{-2}\).

Step 4: Evaluate the Expression

A negative exponent means the reciprocal of the base raised to the positive exponent. So, \(12^{-2}\) is equivalent to \(\frac{1}{12^2}\). Calculate \(12^2\), which is 144. Therefore, \(\frac{1}{144}\) is the final answer.

Key Concepts

Substitute Values

Substituting values into an expression is the first step in evaluating algebraic expressions. It involves replacing variables within an expression with their given numerical values. When you substitute values, you are transitioning from a general expression to a specific one.
For instance, in the exercise provided, we substitute:

• Replace \(x\) with 3
• Replace \(y\) with 4
• Replace \(z\) with -1

This turns \((xy)^{2z}\) into \((3 \times 4)^{2(-1)}\).
Substitution is essential because it simplifies the expression, allowing us to evaluate the numerical result more easily.

Simplify Exponents

After substituting values into the expression, it is important to simplify any exponents present in the expression. Exponents provide a compact way to indicate that a number should be multiplied by itself a certain number of times.
In our example:

• Inside the expression \((3 \times 4)^{2(-1)}\), we first simplify the operation inside the parentheses, resulting in \(12\).

Next, it's time to handle the exponent \(2(-1)\).
Simplifying the exponent involves straightforward multiplication: \(2 \times (-1) = -2\). This helps us rewrite the expression as \(12^{-2}\).
Understanding how to simplify the exponents allows further simplification of the expression.

Expressions with Exponents

Expressions with exponents, such as the ones in our exercise, feature numbers that are raised to a specific power. This operation is a crucial part of algebra and involves repeatedly multiplying a number by itself.
In the expression \(12^{-2}\), we are dealing with an exponent of -2. A negative exponent indicates that instead of multiplying the base by itself, we take the reciprocal and then raise it to a positive power.
When handling positive exponents, you would simply multiply the base by itself the number of times indicated by the exponent. Understanding expressions with exponents helps in simplifying and solving various algebraic equations.

Reciprocal of a Number

The reciprocal of a number is a concept that turns division into multiplication by flipping a fraction. For a number \(a\), its reciprocal is \(\frac{1}{a}\).
When dealing with negative exponents, like in the expression \(12^{-2}\), a reciprocal is used to transform the expression. Specifically, \(12^{-2}\) translates into the reciprocal form \(\frac{1}{12^2}\).
After calculating, we find that \(12^2 = 144\).
Hence:

• \(12^{-2}\) becomes \(\frac{1}{144}\)

Understanding the reciprocal is vital not only in this exercise but also in broader mathematical contexts where division or fraction manipulation is involved.