Problem 2 - Algebra 2

Question

Simplify. \(\left(5 a b^{2}-4 a b+7 a^{2} b\right)(a b)^{-1}\)

Step-by-Step Solution

Answer

The simplified expression is \(5b - 4 + 7a\).

Step 1: Rewrite the Expression

The initial expression is \((5ab^2 - 4ab + 7a^2b)(ab)^{-1}\). We need to simplify this by removing the inverse term. To do this, we will distribute the inverse term \((ab)^{-1}\) to each term inside the parentheses.

Step 2: Distribute the Inverse to Each Term

Distribute \((ab)^{-1}\) to each term inside the parentheses: 1. \((5ab^2)(ab)^{-1}\)2. \(( - 4ab)(ab)^{-1}\)3. \((7a^2b)(ab)^{-1}\).

Step 3: Simplify Each Term

1. For \((5ab^2)(ab)^{-1}\):\[5ab^2 imes (ab)^{-1} = 5\cdot \frac{ab^2}{ab} = 5b\]. 2. For \((-4ab)(ab)^{-1}\):\[-4ab imes (ab)^{-1} = -4 \cdot \frac{ab}{ab} = -4\]. 3. For \((7a^2b)(ab)^{-1}\):\[7a^2b \times (ab)^{-1} = 7 \cdot \frac{a^2b}{ab} = 7a\].

Step 4: Combine the Simplified Terms

Combine the simplified terms:\[5b - 4 + 7a\]. This is the result after simplifying each part of the expression.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions by multiplying a term outside the parentheses by each term inside the parentheses. In this exercise, we're using the distributive property with a twist—the inverse form.
When faced with an expression like equation: \((5ab^2 - 4ab + 7a^2b)(ab)^{-1}\),we apply the distributive property by distributing equation: \((ab)^{-1}\)to every term within the parentheses.
The process involves multiplying each component of the polynomial by the inverse. This operation helps simplify and break down complex expressions into more manageable terms.

• Multiply each term individually by the factor outside the parentheses.
• Ensure that you apply the inverse correctly, transforming conditions like division inside polynomials.

Understanding this property not only helps simplify equations but also strengthens problem-solving skills in algebra.

Exponent Rules

Exponent rules are crucial in managing and simplifying expressions with powers. In this exercise, exponents play a significant role. Here's a brief overview of the rules applied:
When simplifying terms involving powers such as equation: \((ab^2)(ab)^{-1}\),the law of exponents states that when you divide terms with the same base, you subtract the exponents.
For example:

• In the term equation: \(5ab^2 \times (ab)^{-1}\), the base \(ab\) is in both the term and the inverse. The exponent of \(b\) simplifies from 2 to 1 (2 - 1 = 1).
• Similarly, in the division equation: \((a^2b)(ab)^{-1}\), subtract the exponents of \(a\) and \(b\) respectively, yielding \(a^1\).

With these rules, we efficiently reduce terms, ensuring each base is handled correctly. Exponent rules are essential to tackling polynomial expressions and breaking them down into their simplest form.

Algebraic Expressions

An algebraic expression is a mathematical phrase involving numbers, variables, and operations. They're central to algebra, and in this specific problem, we aim to simplify such an expression.
The original expression equation: \(5ab^2 - 4ab + 7a^2b\)is composed of terms involving variables \(a\) and \(b\) with coefficients.
Here's a quick breakdown:

• Each part of the expression is a "term," such as equation: \(5ab^2\).
• Terms like \(-4ab\) represent linear combinations of the included variables \(a\) and \(b\) with their coefficients.
• The expression as a whole is a polynomial, demonstrating how these terms interact with one another through addition and subtraction.

When simplifying, always look at how each term can be reduced or combined with others, keeping in mind the properties of addition and multiplication. This improves both computational efficiency and clarity in algebra.