Problem 9 - Elementary and Intermediate Algebra

Question

Complete each solution. $$ \left(9 n^{3}\right)\left(8 n^{2}\right)=(9 \cdot \quad)\left(\quad \cdot n^{2}\right)= $$

Step-by-Step Solution

Answer

The complete expression is \( 72n^5 \).

Step 1: Recognize the structure

We need to multiply two expressions, \( (9n^3) \) and \( (8n^2) \). Notice that the expressions are in the form of coefficients multiplied by powers of \( n \).

Step 2: Multiply the coefficients

Start by multiplying the numerical coefficients: \( 9 \) and \( 8 \). The result of this multiplication is \( 72 \).

Step 3: Add the exponents of n

Next, handle the powers of \( n \). Since the expression has \( n^3 \) and \( n^2 \), add the exponents: \( 3 + 2 = 5 \).

Step 4: Combine the results

Combine the results from Steps 2 and 3. The complete expression is thus \( 72n^5 \).

Key Concepts

Exponents in Algebra

Exponents are a key aspect of algebraic multiplication, serving as a shorthand notation for repeated multiplication. In the expression \( n^3 \), the exponent \( 3 \) indicates that the base \( n \) is multiplied by itself three times, i.e., \( n \times n \times n \). When dealing with exponents, one of the key rules is that when you multiply powers with the same base, you add their exponents. For example, when multiplying \( n^3 \) and \( n^2 \), you get \( n^{3+2} \), which simplifies to \( n^5 \). This rule helps simplify expressions and makes calculations quicker and more straightforward. Remembering this rule will help you effectively tackle problems involving exponents in polynomials.

Understanding Coefficients

In any algebraic expression, coefficients are the numerical factors that precede the variables. For example, in \( 9n^3 \), the number \( 9 \) is the coefficient. It tells us how many times the base, in this case, \( n^3 \), is being multiplied. When performing operations like multiplication on expressions such as \( 9n^3 \) and \( 8n^2 \), focus on coefficients first. Here, you multiply the coefficients \( 9 \) and \( 8 \) together to get \( 72 \). Once you have the product of the coefficients, you then apply the rules of exponents to the variable part. This step-by-step approach simplifies the handling of polynomial multiplication.

Polynomial Expressions Simplified

A polynomial is a type of mathematical expression that consists of variables and coefficients, often involving operations of addition, subtraction, and multiplication. Each term in a polynomial consists of a coefficient and a variable raised to an exponent, like \( 9n^3 \). When multiplying polynomial expressions, such as \( (9n^3)(8n^2) \), the process involves both coefficients and exponents. First, multiply the coefficients to get \( 72 \). Next, apply the exponent rules to combine the powers of \( n \), resulting in \( n^5 \). The final expression, \( 72n^5 \), represents the simplified form of the original multiplication. Understanding how to manipulate polynomial expressions by handling coefficients and exponents separately can significantly ease your work in algebra.