Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: 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

First, identify and multiply the numerical coefficients in the expression. The coefficients are \(\frac{1}{6}\), \(-3\), and \(4\). Multiply them: \[ \frac{1}{6} \times (-3) \times 4 = \frac{1}{6} \times (-12) = -2. \]

Step 2: Apply the Product of Powers Property

Next, apply the product of powers property to the variable part, \(a^5\), \(a^2\), and \(a^7\). According to this property, when multiplying bases with the same exponent, you add the exponents: \[ a^5 \times a^2 \times a^7 = a^{5+2+7} = a^{14}. \]

Step 3: Combine the Results

Combine the results from Steps 1 and 2 to write the simplified expression. Use the coefficient \(-2\) and the power of \(a\) that we calculated: \[-2a^{14}.\] This is the simplified form of the expression.

Key Concepts

Simplifying Expressions

One of the key skills in algebra is simplifying expressions. When we simplify an expression, we aim to make it as straightforward and concise as possible. This often involves combining numbers and variables in a way that is clearer and more manageable.
Consider the expression \(\left(\frac{1}{6} a^{5}\right)\left(-3 a^{2}\right)\left(4 a^{7}\right)\). To simplify it, we separate the numerical coefficients (\(\frac{1}{6}\), \(-3\), and \(4\)) from the variable parts.

• First, multiply the coefficients. It helps to perform each multiplication step-by-step. Here, we multiply \(\frac{1}{6} \times (-3) \times 4\) together to get \(-2\).
• Then, handle the variable, which is multiple terms of \(a\) with different exponents.

Simplifying expressions requires clear organization and precision, especially when dealing with both coefficients and variables.

Product of Powers Property

A fundamental property used in algebra, the Product of Powers Property, helps in simplifying expressions involving exponents. This property states that when you multiply two powers with the same base, you simply add the exponents.
For example:

• If you have \(a^5\), \(a^2\), and \(a^7\), they all share the base \(a\).
• By using the property, \(a^5 \times a^2 \times a^7\) becomes \(a^{5+2+7} = a^{14}\).

This makes calculations more efficient, especially with large numbers. The Product of Powers Property is invaluable when you are dealing with polynomials or any expressions with exponents because it greatly simplifies the multiplication process.

Polynomials

Polynomials are expressions that include variables raised to various powers and coefficients. In algebra, understanding how to manipulate and simplify polynomials is essential. They can vary from simple expressions like \(3x+7\) to more complex forms such as \(x^3 + 2x^2 - x + 5\).
In the exercise \(\left(\frac{1}{6} a^{5}\right)\left(-3 a^{2}\right)\left(4 a^{7}\right)\), we have a polynomial expression with one type of variable, \(a\). The key operations involve:

• Multiplying coefficients to handle the numerical portion.
• Applying properties like the Product of Powers to simplify the variable part.

These skills allow us to transform a more complex polynomial into a simplified form \(-2a^{14}\), making it easier to use in further calculations or applications. Understanding how to work with polynomials is foundational in algebra and forms the basis for many other mathematical concepts.