Problem 14 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \left(-3 b^{3} c\right)\left(7 b^{2} c^{2}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-21 b^5 c^3\).

Step 1: Distribute the Terms

Start by distributing each component in the first parenthesis across the second parenthesis: \((-3 b^3 c) imes (7 b^2 c^2)\). This gives us: \((-3 imes 7) imes (b^3 imes b^2) imes (c imes c^2)\).

Step 2: Multiply the Coefficients

Multiply the numerical coefficients: \(-3 \times 7 = -21\). This simplifies our expression to: \(-21 \times b^3 \times b^2 \times c \times c^2\).

Step 3: Apply the Product of Powers Property to Variables

Combine the powers of the same base by adding their exponents. For \(b\): - Use \(b^3 \times b^2 = b^{3+2} = b^5\).For \(c\): - Use \(c^1 \times c^2 = c^{1+2} = c^3\).

Step 4: Simplify the Entire Expression

Substitute back the simplified expressions for \(b\) and \(c\) into the product with the numerical coefficient: - The final simplified expression is: \(-21 b^5 c^3\).

Key Concepts

Product of Powers Property

Understanding the Product of Powers Property can greatly simplify working with exponents. This property states that when you multiply two expressions with the same base, you can simply add their exponents. For example, if you have \(b^3 \times b^2\), both terms have the base \(b\). All you need to do is add the exponents: \(3 + 2\). This simplifies to \(b^5\).
This property is incredibly useful for simplifying expressions with variables, as it reduces the need for lengthy multiplication. It also ensures that you end up with a single term for each base, making the final expression much easier to read and interpret.

Exponents

Exponents are a way to represent repeated multiplication of the same number or variable. For example, \(b^3\) means \(b \times b \times b\). Exponents have rules that make working with them easier.
Some common rules are:

• Product of Powers: Add exponents when multiplying like bases, e.g., \(b^3 \times b^2 = b^{3+2} = b^5\).
• Power of a Power: Multiply exponents when raising an exponent to another power, e.g., \((b^3)^2 = b^{3\times2} = b^6\).
• Power of a Product: Apply the exponent to each factor inside the parentheses, e.g., \((ab)^2 = a^2b^2\).

Understanding these rules is crucial in simplifying algebraic expressions effectively.

Coefficients

In an algebraic expression, the coefficient is the numerical part that multiplies a variable. In the expression \(-3b^3\), \(-3\) is the coefficient of \(b^3\). Coefficients can be positive or negative numbers and affect the overall value of the expression.
When multiplying coefficients from different terms, treat them as you would regular numbers: simply multiply them together. For example, when multiplying \(-3\) by \(7\) in the expression \((-3b^3c)\times(7b^2c^2)\), the coefficients multiply to \(-21\). This result then forms the coefficient of the simplified expression \(-21b^5c^3\). Understanding coefficients helps you correctly scale terms when simplifying expressions.

Algebraic Expressions

Algebraic expressions are a mix of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They can represent anything from a simple expression like \(x+5\) to more complex forms like \(-3b^3 + 7b^2\).
When simplifying algebraic expressions, such as \((-3b^3c)\times(7b^2c^2)\), it's crucial to follow the order of operations and apply the properties of exponents and coefficients correctly. This process usually involves simplifying like terms, using the product of powers property, and combining numeric coefficients.
Mastering the simplification of algebraic expressions allows you to solve complex mathematical problems more efficiently and is a fundamental skill in algebra.