Problem 59 - Algebra 1: Concepts and Skills

Question

Simplify the expression. \(\left(a b c^{2}\right)^{3} \cdot a b\)

Step-by-Step Solution

Answer

The simplified expression is \(a^{4}b^{4}c^{6}\).

Step 1: Analyze the Expression

We have the expression \((abc^{2})^{3} \cdot ab\). In this expression, \(abc^{2}\) is raised to the power 3, and then the result is multiplied by \(ab\).

Step 2: Apply the Power of a Product Rule

The power of a product rule states that \((ab)^{n} = a^{n}b^{n}\). In this case, we apply this rule to \((abc^{2})^{3}\) to get \(a^{3}b^{3}(c^{2})^{3}\).

Step 3: Simplify exponents

We can simplify the expression from step 2 to \(a^{3}b^{3}c^{6}\) using the rule of powers \((a^{n})^{m} = a^{n \cdot m}\), where the exponents are multiplied. The product of 2 and 3 is 6.

Step 4: Multiplication

The final step is to multiply by the remaining \(ab\), giving us the simplified expression \(a^{4}b^{4}c^{6}\). In multiplication of terms with exponents, the exponents are added. Therefore, \(a^{3}a^{1} = a^{4}\) and \(b^{3}b^{1} = b^{4}\).

Key Concepts

Power of a Product Rule

Understanding the power of a product rule is essential when working with algebraic expressions involving exponents. This rule simplifies the process of raising a product to an exponent by allowing us to distribute the exponent to each factor within the parentheses.

Let's consider an expression in the form \( (ab)^n \). According to the power of a product rule, we can rewrite this as \( a^n b^n \), which means we separately raise each factor inside the parentheses to the power outside. It's important to apply this rule correctly, as it simplifies more complex expressions and prepares them for further manipulation.

• First, identify the products within parentheses.
• Then, raise each factor of the product to the given power.

By using this rule, we can turn seemingly intimidating expressions into manageable ones, ready for additional simplification.

Simplifying Exponents

Exponents represent repeated multiplication, and simplifying exponents is a fundamental skill in algebra that streamlines expressions for easier understanding and calculation. When an expression with a power is raised to another power, such as \( (a^n)^m \), we multiply the exponents to simplify: \( a^{n \cdot m} \).

• Remember, \( a^n \) represents \( a \times a \times a \)... (n times).
• When raising \( a^n \) to another power \( m \) \( (a^n)^m \), just multiply the exponents \( n \) and \( m \).

This technique helps in consolidating the expression into its most compact form, especially before engaging in operations such as addition or multiplication with other terms.

Multiplying Polynomial Expressions

Multiplying polynomial expressions is a more advanced extension of working with single-variable terms, which is only complicated by the fact that there are more terms involved.

When multiplying polynomials, we use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. The key steps include:

• Identify like terms - terms that have the same variables raised to the same power.
• Multiply coefficients and add the exponents of like terms.
• Combine like terms to simplify the expression.

It's a bit like a dance, where each term in the first polynomial takes a turn at partnering with each term in the second. By following this systematic approach, we can multiply polynomials efficiently and accurately, setting a strong foundation for further algebraic manipulation.