Problem 23 - Elementary and Intermediate Algebra

Question

Multiply. See Example 1. $$ \left(8 a^{5}\right)\left(-\frac{1}{4} a^{6}\right) $$

Step-by-Step Solution

Answer

The result of multiplying \((8a^5)\) and \((-\frac{1}{4}a^6)\) is \(-2a^{11}\).

Step 1: Identify the Components

First, identify the parts inside the given expression to understand what is being multiplied. You have two terms: \(8a^{5}\) and \(-\frac{1}{4}a^{6}\). These contain coefficients (8 and \(-\frac{1}{4}\)) and variables with exponents \(a^{5}\) and \(a^{6}\).

Step 2: Multiply the Coefficients

Multiply the numerical coefficients of the two terms. Simplify: \(8\times -\frac{1}{4}\). This equals \(-2\) (since \(8 \times -\frac{1}{4} = -2\)).

Step 3: Apply the Product Rule to Exponents

Multiply the variables with exponents using the product rule for exponents, which states that \(a^{m} \times a^{n} = a^{m+n}\). For the expression, multiply \(a^{5}\) and \(a^{6}\) as follows: \(a^{5+6} = a^{11}\).

Step 4: Combine the Results

Now, combine the results from Steps 2 and 3. The product of the coefficients is \(-2\) and the product of the variables is \(a^{11}\), giving the final expression: \(-2a^{11}\).

Key Concepts

Coefficients

When dealing with algebraic multiplication, coefficients play a crucial role. In mathematical expressions, a coefficient is the numerical factor that multiplies a variable. In the expression \(8a^5\), the number 8 is the coefficient of the variable \(a^5\). Likewise, in \(-\frac{1}{4}a^6\), the coefficient is \(-\frac{1}{4}\). This shows the constant proportionality each term has to the variable involved.When multiplying two expressions, you start by multiplying their coefficients. This is done just like you would in regular arithmetic. For instance, to multiply the coefficients 8 and \(-\frac{1}{4}\), you simply perform the multiplication:

• 8 times \(-\frac{1}{4}\) equals \(-2\).

This gives you a simplified coefficient in the resulting expression.It's important to remember that coefficients can be positive or negative, affecting the sign of the product. Multiplying a positive number by a negative number always results in a negative product, as seen in our example.

Exponents

In algebra, exponents denote how many times a number, known as the base, is multiplied by itself. When considering the expression \(a^5\), the 5 is the exponent, and it tells us that the base \(a\) is used as a factor five times in multiplication. Similarly, the term \(a^6\) means that \(a\) is repeated six times as a factor.Exponents are particularly useful when multiplying terms with the same base. This is because there is a specific rule that governs their multiplication:

• When you multiply two powers with the same base, you add their exponents.

This is known as the product rule for exponents. Applying it allows you to concisely combine larger expressions.For example, with \(a^5\) and \(a^6\), you apply the product rule:

• The exponents are added: \(5 + 6 = 11\).

Thus, the expression \(a^5 \times a^6\) simplifies to \(a^{11}\). This keeps expressions more manageable and ensures you maintain the integrity of the base and its operations.

Product Rule

The product rule for exponents simplifies the process of multiplying exponential terms. It is an essential algebraic rule that states: when multiplying two powers with the same base, you add the exponents. This rule applies when you encounter terms like \(a^m \times a^n\).Here's how it works:

• Assume you have \(a^5\) and \(a^6\).
• The product rule says you should add the exponents: \(5 + 6\).
• The result is \(a^{11}\), which summarizes the original multiplication in a compact form.

Using the product rule not only simplifies expressions but also helps maintain clarity when working with polynomial expressions. This rule is particularly useful in reducing complex algebraic problems to more straightforward expressions, allowing for easier computation and interpretation.