Problem 45 - Beginning and Intermediate Algebra

Question

Simplify. $$\left(3 m^{4}\right)^{3}\left(2 m^{-5}\right)^{4}$$

Step-by-Step Solution

Answer

The simplified expression is \(432m^{-8}\).

Step 1: Apply Exponent Rules to Each Term Individually

We will first apply the exponent rules to each term separately. Recall that the exponent rule \((a^m)^n = a^{mn}\). For the first term, \((3m^4)^3\), we have: \[(3m^4)^3 = 3^3 \cdot m^{4\cdot 3} = 27m^{12}\]. For the second term, \((2m^{-5})^4\), we have: \[(2m^{-5})^4 = 2^4 \cdot m^{-5\cdot 4} = 16m^{-20}\].

Step 2: Multiply the Simplified Terms

Now, we will multiply the simplified terms \((27m^{12})(16m^{-20})\). We can multiply the constants and the variable part separately. The constants are 27 and 16. Multiplying them, we have: \[27 \cdot 16 = 432\]. The variable part is \(m^{12}\) and \(m^{-20}\). Using the exponent rule \(a^m\cdot a^n = a^{m+n}\), we have: \[m^{12}\cdot m^{-20} = m^{12 - 20} = m^{-8}\].

Step 3: Combine the Result

Now, we will combine the result from the multiplication to get the final simplified expression: \[432 \cdot m^{-8}\]. The simplified expression is \(432m^{-8}\).

Key Concepts

Simplifying Expressions

When simplifying expressions, the goal is to reduce them to their simplest form. This often involves using algebraic rules to combine like terms and eliminate complex parts. For the given exercise, we're simplifying an expression with exponents.

The key here is understanding the notation and applying the rules of exponents correctly. With exponentiation, especially when variables and coefficients are involved, it's important to follow:

• Each part of the expression should be evaluated separately.
• Look out for common factors or terms that can be combined.

In this particular problem, the expression \((3m^4)^3(2m^{-5})^4\) involves both positive and negative exponents, which means it's not just the variables but also the numerical parts that need careful attention. Simplifying complex expressions requires practice, allowing for the consistent application of rules and ensuring no step is overlooked. Once the expression is entirely broken down, it can be rebuilt in its simplest, most comprehensible form.

Negative Exponents

Negative exponents often cause confusion, but they are quite simple once you grasp the concept. A negative exponent indicates that the base of the expression is on the opposite side of a fraction.

In other words, for any number or variable, \(a^{-n} = \frac{1}{a^n}\). This means it's not really negative but represents a reciprocal or a fraction.

• Negative exponents make the base move to the denominator of a fraction.
• When combined with multiplication, these terms need careful handling.

In our problem, \(m^{-20}\) was combined with \(m^{12}\). The exponent rule for multiplying like bases directs us to add exponents, resulting in a new expression with \(m^{12 - 20} = m^{-8}\). Recognizing negative exponents as reciprocals can help you better understand how to handle them during simplification and multiplication.

Multiplying Like Bases

Multiplying like bases, such as in \(m^{12} \cdot m^{-20}\), requires using exponent rules that make the process simple and efficient. When you have the same base, you simply add the exponents.

Why is this important? Adding exponents keeps the operation straightforward and maintains the integrity of the variable.

• The base remains the same; only the exponents are affected.
• Remember, the product rule states \(a^m \cdot a^n = a^{m+n}\).

In our example, when multiplying \(m^{12}\) and \(m^{-20}\), we applied the product rule: \(m^{12 + (-20)} = m^{-8}\). This demonstrates how exponent rules consolidate terms without changing the base.This approach works similarly with any like bases, ensuring calculations remain manageable, even as expressions become complex.