Problem 93 - Algebra 1

Question

SIMPLIFYING EXPRESSIONS Simplify. Write your answer as a power or as an expression containing powers. $$6 x \cdot(6 x)^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(216x^3\).

Step 1: Understand the expression.

The expression to simplify is \(6x \cdot (6x)^2\). Here, \(6x\) is multiplied by \((6x)^2\). Note that \(6x\) can also be written as \((6x)^1\).

Step 2: Multiply the similar terms.

According to the exponent rules, when we multiply similar terms, we add the exponents. Hence, the expression becomes \((6x)^{1+2} = (6x)^3\).

Step 3: Simplify the expression.

The expression \((6x)^3\) denotes that \(6x\) should be multiplied by itself three times. Therefore, the expression simplifies to \(216x^3\).

Key Concepts

Exponent Rules

Exponent rules are simple mathematical guidelines that help simplify expressions involving powers and exponents. They are crucial for tasks like our original exercise, as they give us a systematic way to approach complex expressions. One key rule is the "Product of Powers" rule. This rule tells us that when we multiply expressions with the same base, we keep the base and add the exponents together. For example, if you have

• \(a^m \cdot a^n\), the result is \(a^{m+n}\).

This simplifies calculations and reduces expressions to a more manageable form. In the exercise, applying this rule helps us move from

• \(6x \cdot (6x)^2\) to \((6x)^{1+2}\).

Understanding these rules is not just about memorizing formulas. It's about recognizing patterns and relationships in numbers. Once you see how exponents work, simplifying expressions becomes a breeze.

Multiplying Powers

Multiplying powers involves applying exponent rules to combine terms into a single power expression. It's a vital step in expression simplification. When dealing with powers, the process can be broken down as follows:

• Identify terms with the same base.
• Use the "Product of Powers" rule to add their exponents.

In our example exercise, multiplying

• \(6x \cdot (6x)^2\)

means recognizing that both terms have the base

• \(6x\)

and adding their exponents,

• \(1+2=3\).

This process transforms the expression into a more straightforward form,

• \((6x)^3\).

The beauty of multiplying powers is that it simplifies not only numbers but also the calculations, as seen in our example.

Expression Simplification

Expression simplification is all about breaking down a complex mathematical expression into its simplest form. It harnesses the power of exponent rules and multiplying powers to achieve this goal. To simplify an expression like \((6x)^3\), you must compute the power, meaning you multiply the base

• \(6x\) by itself three times: \(6x \cdot 6x \cdot 6x\).

First, focus on the numerical part:

• \(6 \cdot 6 \cdot 6 = 216\).

Next, handle the variable's power by multiplying the coefficient and keeping the exponent:

• \(x^1 \cdot x^1 \cdot x^1 = x^3\).

The final simplified form in our original problem,

• \(216x^3\),

shows how structure and understanding transform complex expressions into simpler, more understandable results. Simplification makes it easier to solve problems and spot errors if they arise.