Problem 93 - Algebra A Combined Function

Question

Simplify each expression. $$ (9 x y)^{2} $$

Step-by-Step Solution

Answer

\( (9xy)^{2} = 81x^{2}y^{2} \)

Step 1: Understand the Problem

We are tasked with simplifying the expression \((9 x y)^{2}\). This involves applying the rules of exponents to simplify the expression in its simplest form.

Step 2: Apply the Power of a Product Rule

The power of a product rule states that \((a \, b)^{n} = a^{n} \, b^{n}\). Here, we can rewrite \((9xy)^{2}\) as \(9^{2} \cdot (xy)^{2}\).

Step 3: Simplify Each Factor

Now, simplify each separate part. First, calculate the square of 9: \(9^{2} = 81\). Then handle the remaining \((xy)^{2}\) part using the power of a product rule again: \((xy)^{2} = x^{2} \, y^{2}\).

Step 4: Combine the Simplified Parts

Combine the simplified parts from the previous step. Thus, \((9xy)^{2} = 81 \cdot x^{2} \cdot y^{2}\).

Step 5: Write the Simplified Expression

Put it all together to express the simplification: \((9xy)^{2} = 81x^{2}y^{2}\).

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of the same number or variable. They are incredibly useful in mathematics for simplifying expressions and solving equations. When we talk about exponents, the number that is being multiplied is known as the base, and the number of times it is multiplied by itself is the exponent.

For example, in the expression \[a^b\]- **Base (a):** The number being multiplied
- **Exponent (b):** The number that expresses how many times the base is multiplied by itself

In the context of the exercise, \[(9xy)^2\]means the expression is being multiplied by itself once. Exponents help in writing this repeated multiplication in a compact form.

Power of a Product Rule

The power of a product rule is a fundamental concept when dealing with exponents, especially in expressions that include multiple terms. This rule states:\[(a \cdot b)^n = a^n \cdot b^n\]This means that when you have a product inside the parentheses raised to an exponent, you can distribute the exponent to each factor inside the parentheses separately.

In the exercise, the expression \[(9xy)^2\]is simplified using the power of a product rule. Here's how it works step by step:

• Take the product inside the parentheses: 9, x, and y.
• Raise each component to the power of 2: \(9^2\), \(x^2\), and \(y^2\).

So, \[(9xy)^2\]simplifies to \[9^2\cdot x^2\cdot y^2\].

Simplifying Expressions

Simplifying expressions is the process of making them easier to work with by following certain mathematical rules. This often involves combining like terms, using exponent rules, and performing arithmetic operations to make the expression as concise as possible.

In the given exercise, after applying the power of a product rule, we proceed with actually calculating the exponents:

• First, compute \(9^2\). Since \(9 \times 9 = 81\), we have 81.
• Next, apply the power to each variable separately: \(x^2\) and \(y^2\).

Finally, the simplified expression \[81x^2y^2\]is obtained. This version is simpler and easier to interpret or use in further calculations. Simplifying expressions like these is crucial for solving algebraic equations and simplifying complex algebraic tasks.