Problem 28 - Algebra and Trigonometry

Question

\(25-28\) . Evaluate the expression using \(x=3, y=4,\) and \(z=-1\) $$ (x y)^{2} $$

Step-by-Step Solution

Answer

144

Step 1: Substitute the values

First, substitute the given values of the variables into the expression. The expression is \((xy)^2\). With \(x = 3\) and \(y = 4\), substitute to get \((3 \times 4)^2\).

Step 2: Simplify inside the parentheses

Simplify the product inside the parentheses. Calculate \(3 \times 4\) to get \(12\). The expression is now \(12^2\).

Step 3: Evaluate the exponent

Square the value obtained in the previous step. Calculate \(12^2 = 144\).

Key Concepts

Substitution in Algebra

Substitution in algebra is the process where you replace variables in an expression with their given numerical values. This technique is essential when you need to evaluate algebraic expressions. It makes them straightforward numerical calculations.
For example, in the expression \((xy)^2\), once we're given specific values for \(x\) and \(y\), such as \(x = 3\) and \(y = 4\), we can directly substitute these numbers into the expression.
Here’s how you do it:

• Identify each variable and its corresponding value.
• Replace each variable in the expression with its specified number.
• Rearrange and simplify the expression, if needed, to make calculations easier.

Substitution bridges the gap between abstract algebraic concepts and concrete numerical results.

Simplifying Expressions

Simplifying expressions is a process of rewriting them in an easier or more compact form without changing their value. This often involves combining like terms and performing operations specified within parenthesis.
Once substitution has been done, the next task is typically to simplify what remains. In our example, after substituting \(x = 3\) and \(y = 4\) into \((xy)^2\), you obtain \((3 \times 4)^2\).
Here's how the simplification process works:

• Perform any operations inside parentheses first. For example, calculate \(3 \times 4\) to get \(12\).
• Ensure the expression is at its simplest form. In our case, the expression becomes \(12^2\).

This logical sequence helps break the problem into manageable steps and is crucial for correct calculations.

Exponentiation

Exponentiation is a mathematical operation involving the raising of a number, known as the base, to a power, which is called the exponent. The exponent tells us how many times to multiply the base by itself.
In the expression \((xy)^2\), after substitution and simplification, you end up with \(12^2\). This means multiplying 12 by itself: \(12 \times 12\).
Here are key points to remember about exponentiation:

• The notation \(a^n\) indicates 'a' to the power of 'n'.
• An exponent of 2 is known as squaring. It results in \(a \times a\).
• Always apply exponents after dealing with inside-parentheses simplifications.

By mastering exponentiation, you unlock the ability to efficiently handle complex expressions, a crucial skill in algebra.