Problem 35 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ (3 b)^{3} \cdot b $$

Step-by-Step Solution

Answer

The expression simplifies to \(27b^4\).

Step 1: Break Down the Expression

First, break down \((3b)^3\) into \(3^3 * b^3\), resulting in \(27b^3\). The given problem becomes \(27b^3 * b\).

Step 2: Apply the Multiplication Rule

In the second step, apply the multiplication rule for expressions with the same base; when the base is the same (in this case, \(b\)), we add the exponents. So, \(b^3 * b\) can be rewritten as \(b^{3+1}\), which equals to \(b^4\). The expression now looks like \(27*b^4\).

Step 3: Final Simplification

The final step simply involves writing the final answer. Here, \(27*b^4\) remains the same as the expression is already simplified.

Key Concepts

Multiplication Rule

The Multiplication Rule is a fundamental concept in algebra, especially when dealing with exponents. It states that when you multiply two expressions with the same base, you can simply add the exponents together. Consider the example from our exercise where we have

• \(b^3 * b\)

Here, both terms share the base \(b\).
By applying the multiplication rule, we add the exponents: \(3 + 1\).
This gives us \(b^4\).
It's important to remember that this rule only applies when the bases are identical; otherwise, you cannot combine the exponents in such a way.

Simplified Expression

A simplified expression is one that has been reduced to the simplest possible form, with no further operations that can make it shorter or clearer. In the context of the exercise, after breaking down the initial expression and applying the multiplication rule, we obtain

• \(27*b^4\)

This is our simplified expression.
Originally, we had an expression:

• \((3b)^{3} \cdot b\),

which involved multiple terms and operations, but through systematic reduction, it has been converted to a form that is concise and easy to understand.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations, forming a sort of mathematical phrase that's key in algebra. In our exercise,

• \((3b)^{3} \cdot b\)

serves as a typical algebraic expression. It includes the constant 3, the variable \(b\), and operations such as exponentiation and multiplication.
When working with algebraic expressions, it's crucial to understand the roles of different components.
Constants like the number 3 provide a fixed reference, while variables like \(b\) represent unknown values that can change.
Operations guide how these elements are combined or changed to form new expressions. Mastery of handling these components helps in efficiently solving algebraic problems.