Problem 84 - Intermediate Algebra

Question

Simplify each expression. $$ b+\frac{b}{1-\frac{b+1}{b}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-b^2 + b\).

Step 1: Simplify the Inner Fraction

The expression inside the denominator is: \[ 1 - \frac{b+1}{b} \]Start by writing it as a single fraction.Expand the 1 to \( \frac{b}{b} \). This gives us:\[ \frac{b}{b} - \frac{b+1}{b} \]Since the denominators are the same, we can subtract them:\[ \frac{b - (b+1)}{b} = \frac{b - b - 1}{b} = \frac{-1}{b} \]

Step 2: Simplify the Overall Denominator

Now replace the denominator in the original expression with the result from Step 1:\[ b + \frac{b}{\frac{-1}{b}} \]The \( \frac{b}{\frac{-1}{b}} \) can be simplified by multiplying by the reciprocal:\[ \frac{b}{\frac{-1}{b}} = b \times \left(-b\right) = -b^2 \]

Step 3: Combine Terms

Now, the original expression becomes:\[ b - b^2 \]The terms can be combined as they are polynomial terms. So, the simplified expression is:\[ -b^2 + b \]

Key Concepts

Simplifying Expressions

Simplifying expressions means making them easier to work with by reducing them to their most basic form. In a simplified expression, unnecessary elements are removed or rewritten in a way that makes further calculations easier. To simplify, we perform operations like combining like terms and reducing fractions.
In the given problem, simplifying requires handling both the numerator and the denominator. We first focused on the inner fraction. The main goal in this step was to rewrite the expression in a single fraction form, allowing us to subtract and find one simple fraction. For example, changing 1 to \( \frac{b}{b} \) allowed us to simplify \( 1 - \frac{b+1}{b} \) to \( \frac{-1}{b} \). This might seem tricky at first, but breaking it down step-by-step makes it clearer.
Remember, keeping track of minus signs and common denominators ensures accurate simplification.

Polynomial Operations

Polynomial operations involve handling expressions that include multiple terms made by variables raised to powers, typically through addition or subtraction. In our exercise, once the fraction was simplified, we ended up with a polynomial \( b - b^2 \).
When dealing with polynomials, think of each term separately. Here, you have \( b \) and \( -b^2 \). Combining them relies on identifying and dealing with like terms, which are terms that have the same variable to the same power. In our example, there is no like term for \( b^2 \) since there's only one term with an "\( b^2 \)" in it. This means we can't combine \( b \) with \( -b^2 \), and so we leave them as a single expression.
Polynomials need careful handling, especially when operations like subtraction are involved. A small mistake with a sign can mislead the whole problem. Always double-check your work to ensure sign accuracy.

Fraction Simplification

Fraction simplification is a crucial step in algebra, where a fraction is reduced to its simplest form. This means making the numerator and the denominator as basic as possible.
For the given exercise, the challenge was dealing first with the inner fraction \( 1 - \frac{b+1}{b} \) which involved rewriting it into \( \frac{-1}{b} \).
Another essential step was dividing \( b \) by the fraction \( \frac{-1}{b} \), which involved flipping the inner fraction and changing division into multiplication. Calculating \( \frac{b}{\frac{-1}{b}} \) simplifies down to \(-b^2 \) by multiplying \( b \) with \(-b\).

• Recognizing when and how to take the reciprocal of a fraction is a fundamental skill. This step effectively removes the fractions, turning complex expressions into simpler polynomials for further operations.
• Always be meticulous with signs when flipping fractions, as a sign error can change the entire result.

Simple checks for simplification can include verifying whether the numerator and denominator share any common factors, which would allow further reduction.