Problem 12 - Algebra 1: Concepts and Skills

Question

Use the power of a product property to simplify the expression. \((3 p q)^{3}\)

Step-by-Step Solution

Answer

The simplified version of the expression \((3pq)^3\) is \(27p^3q^3\)

Step 1: Apply the power of a product rule

We're going to apply the power of a product rule which states that \((ab)^n = a^n * b^n\). Here, 'a' is 3, 'b' is 'pq' and 'n' is 3 i.e. \((3pq)^3 = 3^3 * (pq)^3\)

Step 2: Simplify further

Now we simplify 3^3 and apply the power property further to term '(pq)^3' resulting in \(27p^3q^3\).

Key Concepts

Simplifying Algebraic Expressions

The process of simplifying algebraic expressions is a fundamental skill in algebra that involves reducing expressions to their simplest form. This often includes combining like terms, removing parentheses, and applying various algebraic properties. In our original exercise, the expression \( (3 p q)^{3} \) initially looks complex, but by understanding how to simplify it, we create a more manageable expression.

To start simplifying, we identify terms that can be manipulated—such as those with exponents or products that can be expanded using algebraic properties like the power of a product property. This specific property allows us to distribute exponents over the terms inside the parentheses, transforming the initial expression into a product of simpler terms, each raised individually to the power of three.

By converting \( (3 p q)^{3} \) into \( 27p^3q^3 \) in our solution, we've successfully simplified the expression, illustrating the utility and necessity of mastering simplification techniques in algebra.

Exponents and Powers

Exponents and powers are shorthand for expressing repeated multiplication. For example, the exponent in \(a^n\) tells us to multiply the base \(a\) by itself \(n\) times. This is an indispensable tool in algebra as it allows for the concise representation and manipulation of large numbers and expressions.

In the exercise \( (3 p q)^{3} \) we see the exponent 3, which instructs us to multiply the product \(3pq\) by itself three times. The power of a product property simplifies this process; instead of carrying out the multiplication, we raise each factor in the product to the third power individually. Thus, both the numerical coefficient (3) and the variables (p and q) are raised to the power of three. This way, heavy computational work is condensed into a quick and efficient procedure, turning \(27 * p^3 * q^3\) into an easily readable and simplified form.

Algebraic Properties

Algebraic properties are the rules that govern the operations in algebra. These include the distributive property, commutative and associative properties, among others. The power of a product property is specific to exponents and is a direct result of the associative property of multiplication.

In our example, applying the power of a product property is much like distributing the exponent over the factors within the parentheses. Specifically, each factor of the product \( (3pq) \) is raised to the power of three, which follows directly from how multiplication is managed under exponentiation, that is multiplying the base repetitively. This illustrates how algebraic properties act as the backbone for simplifying complex expressions and enable us to manipulate and rewrite equations or expressions into more manageable forms without altering their values.