Problem 76 - Algebra A Combined Function

Question

Multiply. $$ 6 r^{3}\left(7 r^{10}\right) $$

Step-by-Step Solution

Answer

The product is \(42r^{13}\).

Step 1: Identify the Problem

We need to multiply two terms: \(6r^3\) and \(7r^{10}\). Both terms involve coefficients and powers of \(r\).

Step 2: Multiply the Coefficients

Multiply the coefficients of the terms. The coefficients are \(6\) and \(7\). Compute \(6 \times 7 = 42\). The result is the coefficient of the final product.

Step 3: Add the Exponents of the Variable

Since we are multiplying terms with the same base \(r\), we add the exponents. The exponents are \(3\) and \(10\). Compute \(3 + 10 = 13\). Thus, the exponent of \(r\) in the product is \(13\).

Step 4: Write the Final Expression

Combine the coefficient from Step 2 and the resulting power of \(r\) from Step 3. The final expression is \(42r^{13}\).

Key Concepts

Coefficients

In algebra, coefficients are the numerical factors that multiply the variable in an algebraic expression. They help to define the magnitude of the term and play a crucial role in operations like addition, subtraction, and multiplication when dealing with expressions and equations.

• In our problem, the coefficients are the numbers 6 and 7.
• When multiplying polynomials, start by multiplying the coefficients of each term.
• For example, in the expression given, the coefficients are multiplied as: \(6 imes 7 = 42\).

This product becomes the new coefficient of the simplified term. Coefficients make it easier to manage the arithmetic when working with algebraic expressions.

Exponents

Exponents indicate how many times a number, known as the base, is multiplied by itself. Understanding how to handle exponents is vital in polynomial multiplication as it directly affects the simplification of terms.

• In our context, the base is the variable, \(r\).
• The exponents associated with \(r\) in our exercise are 3 and 10.
• When multiplying terms with the same base, their exponents are added together. So, we compute: \(3 + 10 = 13\).

This concept simplifies expressions into a single term with a more manageable form. Adding exponents is applied universally across different bases when multiplying, making it a foundational principle in algebra.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operational symbols, such as addition or multiplication, that represent a mathematical relationship or solve a problem.

• In expressions, terms can consist of coefficients (numerical part) and variables raised to a power (the exponent).
• Our expression involves multiplying \(6r^3\) by \(7r^{10}\).
• This process integrates both the multiplication of coefficients and the addition of exponents to form a new, simplified term: \(42r^{13}\).

Understanding how to manipulate these expressions through operations like multiplication is key to solving complex algebraic problems, allowing students to simplify and solve equations efficiently.