Problem 15 - Precalculus: Functions and Graphs

Question

Simplify. $$\left(\frac{1}{6} a^{5}\right)\left(-3 a^{2}\right)\left(4 a^{7}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(-2a^{14}\).

Step 1: Multiply the Coefficients

The expression given is \( \left(\frac{1}{6} a^{5}\right)\left(-3 a^{2}\right)\left(4 a^{7}\right) \). First, focus on simplifying the coefficients: \( \frac{1}{6} \times -3 \times 4 \). Calculate this as \( \frac{1}{6} \times -12 = -2 \). So the coefficients multiply to \( -2 \).

Step 2: Combine the Exponents

Next, combine the exponents of \( a \). Add the exponents from \( a^{5} \), \( a^{2} \), and \( a^{7} \) together by using the property of exponents \( a^{m} \times a^{n} = a^{m+n} \). This gives us \( a^{5+2+7} = a^{14} \).

Step 3: Write the Simplified Expression

Combine the simplified coefficient from Step 1 and the combined exponents from Step 2. The expression becomes \( -2a^{14} \). This is the simplified form of the original expression.

Key Concepts

Exponent Rules

Exponent rules are crucial in simplifying expressions. These rules help us understand how to manipulate powers of numbers and variables effectively. A basic rule used in this exercise is when multiplying like bases, you add their exponents. For instance, when you have terms like \(a^m \times a^n\), the result is \(a^{m+n}\). This rule arises because multiplying exponents is essentially repeated multiplication of the base. So, if you multiply \(a^2\) by \(a^3\), you really have \((a \times a) \times (a \times a \times a)\), which equals \(a^{5}\).
Learning these rules is essential for handling more complex algebraic expressions, especially in polynomial expressions where terms involve variables raised to various powers. Here, understanding how to add exponents allows you to transform the expression into a simplified form effectively.
In our specific example, the terms \(a^5, a^2,\) and \( a^7\) are combined to \(a^{14}\), which is simplified using the rule \( a^m \times a^n = a^{m+n} \). It makes calculations more straightforward, and the final result is much cleaner.

Coefficient Multiplication

Coefficient multiplication is another fundamental concept in algebra. It involves multiplying the numerical parts (coefficients) of algebraic terms separately from the variable parts. This keeps the process organized and manageable.
In this exercise, the coefficients given are \( \frac{1}{6}, -3,\) and \(4\). First, multiply \(\frac{1}{6}\) by \(-3\), which results in \(-\frac{1}{2}\). Then, multiply \(-\frac{1}{2}\) by \(4\), which gives \(-2\). Performing these calculations separately from the exponents ensures clarity and accuracy. This step was crucial to arrive at the final result of \(-2\).
When working with coefficients, always remember to follow the arithmetic rules, such as multiplying fractions and integers, and keeping track of positive and negative signs. Mastery of coefficient multiplication will make handling more complex expressions much easier.

Algebraic Expressions

Algebraic expressions are combinations of numbers and variables connected by operations like addition, subtraction, multiplication, and division. In algebra, we often simplify these expressions to make calculations easier or to solve equations efficiently.
In our exercise, we started with an expression \( \left(\frac{1}{6} a^{5}\right)\left(-3 a^{2}\right)\left(4 a^{7}\right) \) and transformed it into a simpler form. This simplification involved breaking down the problem into manageable parts: simplifying coefficients and combining exponents.
Understanding algebraic expressions involves recognizing the components of each term, knowing how to apply various mathematical operations, and simplifying results when possible. This helps in answering a wide range of mathematical questions, from basic arithmetic to more advanced algebraic problem-solving. When you simplify an expression, you focus on finding the simplest version of a complex equation or formula, resulting in expressions like \(-2a^{14}\). Reading and writing algebraic expressions fluently is fundamental to success in further mathematical studies.