Problem 30 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \left(3 m^{3}\right)^{4} $$

Step-by-Step Solution

Answer

Question: Simplify the expression (3m^3)^4. Answer: The simplified expression is 81m^12.

Step 1: Identify the base and the exponent in the original expression

The expression given is: $$ \left(3 m^{3}\right)^{4} $$ Here, both the factor 3 and the variable \(m^3\) are raised to the power of 4.

Step 2: Apply the power rule for exponents on a product

According to the power rules for exponents, when a product \(ab\) is raised to a power, the result is obtained by raising each factor of the product to the power: $$ (ab)^{n} = a^{n}b^{n} $$ Applying this rule to our expression: $$ \left(3 m^{3}\right)^{4} = 3^4 (m^3)^4 $$

Step 3: Simplify the powers in the expression

Now, we simplify the powers in the expression. We start with the constant 3: $$ 3^4 = 3\times3\times3\times3 = 81 $$ For the variable \(m\), we apply another power rule for exponents: When a power \(a^m\) is raised to another power \(n\), the result is obtained by multiplying the exponents: $$ (a^{m})^{n} = a^{mn} $$ Applying this rule to our variable \(m\): $$ (m^3)^4 = m^{3\times4} =m^{12} $$

Step 4: Write the final simplified expression

Now, we write down the simplified expression by combining the results of step 3: $$ \left(3 m^{3}\right)^{4} = 3^4 (m^3)^4 = 81 m^{12} $$ Therefore, the simplified form of the given expression is: $$ 81m^{12} $$

Key Concepts

Applying Power Rules

Understanding the power rules for exponents is essential when simplifying expressions involving powers. These rules help us manipulate expressions in algebra by giving us a method to handle different exponent-related problems efficiently. For instance:

• **Power of a Product Rule:** If we have a product of two terms, say \(ab\), and this whole product is raised to a power \(n\), we can apply the rule \((ab)^n = a^n b^n\). This involves raising each factor inside the parenthesis to the power separately.
• **Power of a Power Rule:** When a term like \(a^m\) is raised to another power \(n\), it simplifies to \(a^{m \times n}\). Simply multiply the exponents together.

These rules are anchor concepts in algebra and are often the first steps when tackling problems involving exponents. By correctly applying these rules, what seems like a complex expression becomes more manageable. This basic manipulation is at the heart of more advanced mathematical computations.

Simplifying Expressions

The process of simplifying expressions in mathematics involves reducing the expression to its simplest form. This means making it as compact and straightforward as possible without changing its value. When simplifying expressions with exponents, using power rules becomes crucial. Here’s a step-by-step approach:

• **Identify the Components:** Determine which parts of the expression involve products and powers.
• **Apply Power Rules:** Use the appropriate power rules to separate and simplify each component of the expression.
• **Calculate Numerical Powers:** Simplify numerical bases by performing exponentiation, such as calculating \(3^4 = 81\).
• **Combine the Results:** Once all components are simplified, combine them to form the final simplified expression.

By following these steps, we break down a seemingly complicated expression like \((3m^3)^4\) into a more straightforward form \(81m^{12}\). Simplifying expressions not only helps in solving equations but also in understanding the problem more clearly, making further calculations more efficient.

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They serve as the foundation for many problems in algebra and beyond. In dealing with these expressions, specifically those with exponents, understanding the nature and behavior of each component is essential. Considerations to make include:

• **Variables with Exponents:** In expressions like \(m^3\), \(m\) is referred to as the base, while \(3\) is the exponent. The process of raising a variable to a power represents repeated multiplication.
• **Constants and Coefficients:** In the expression \(3m^3\), \(3\) is a coefficient, and it serves as a multiplier for the variable term. Coefficients can also be raised to a power, as seen when \((3m^3)^4\) becomes \(3^4\).
• **Combining Like Terms:** While simplifying, ensure like terms are combined when possible, streamlining the algebraic expression.

Recognizing the structure of algebraic expressions allows for more effective manipulation and simplification, which is critical in both solving equations and applying algebra to real-world problems. Insight into these structures equips students with the tools needed for further mathematical exploration.