Problem 83 - Intermediate Algebra

Question

Simplify each expression. $$ a+\frac{a}{1+\frac{a}{a+1}} $$

Step-by-Step Solution

Answer

\( \frac{3a^2 + 2a}{2a+1} \)

Step 1: Identify the Expression

The given expression is \( a + \frac{a}{1+\frac{a}{a+1}} \). We need to simplify this nested fraction.

Step 2: Simplify the Inner Fraction

The inner fraction is \( \frac{a}{a+1} \). This is already in its simplest form and will be used to simplify the outer expression.

Step 3: Express the Denominator as a Single Fraction

Consider the denominator \( 1 + \frac{a}{a+1} \). This can be rewritten with a common denominator: \( \frac{a+1}{a+1} + \frac{a}{a+1} = \frac{a+1+a}{a+1} = \frac{2a+1}{a+1} \).

Step 4: Simplify the Outer Fraction

Now we simplify \( \frac{a}{\frac{2a+1}{a+1}} \) by multiplying by the reciprocal: \( a \times \frac{a+1}{2a+1} = \frac{a(a+1)}{2a+1} \).

Step 5: Combine with the First Term

The entire expression becomes \( a + \frac{a(a+1)}{2a+1} \). Find a common denominator: \( \frac{a(2a+1)}{2a+1} + \frac{a(a+1)}{2a+1} = \frac{a(2a+1) + a^2 + a}{2a+1} \).

Step 6: Simplify the Numerator

Combine like terms in the numerator: \( 2a^2 + a + a^2 + a = 3a^2 + 2a \). Thus, the expression simplifies to \( \frac{3a^2 + 2a}{2a+1} \).

Key Concepts

Fraction Simplification

Fraction simplification is all about reducing a fraction to its smallest possible form. You aim to have the numerator and the denominator with the least numbers. Here's how you can simplify fractions effectively:

• First, check if the numerator and denominator share any common factors. If they do, divide both by the greatest common factor to simplify the fraction easily.
• Look at the entire fraction to see if both parts are divisible by the same number. Continue dividing until you can no longer do so without using decimals.
• Remember, fractions do not change their value when both the numerator and the denominator are divided by the same non-zero number; they simply represent the same quantity more simply.

In the expression given, fraction simplification plays out when we handled the term \( \frac{a}{1+\frac{a}{a+1}} \) to make it easier to work with. We did this by simplifying the denominator step-by-step until the whole expression was manageable.

Combining Like Terms

Combining like terms is important when you are handling expressions with variables. It's about bringing together terms that have the same variable parts. Here’s how you can think about it:

• Look for terms that have exactly the same variables raised to the same powers. Only these can be combined.
• Sum the coefficients of these like terms. Do not change the variable part during this process.
• This process helps to make complicated expressions shorter and simpler to deal with.

In the last part of the solution, we combined \(2a^2 + a + a^2 + a\) into \(3a^2 + 2a\). By doing this, we were able to make the expression more concise.

Common Denominators

To add or subtract fractions, especially when they have different denominators, you need a common denominator. This is an essential concept that makes working with fractions smooth and straightforward:

• Find the least common multiple of the denominators. This will be your common denominator.
• Adjust each fraction so that both have this common denominator by multiplying the numerator and denominator accordingly.
• Once the fractions have the same denominator, add or subtract the numerators directly.

In our exercise, we used a common denominator when rewriting \( 1 + \frac{a}{a+1} \) to \( \frac{a+1}{a+1} + \frac{a}{a+1}\), which simplified to \(\frac{2a+1}{a+1}\). This step was crucial as it allowed us to simplify the bigger expression without changing its value.

Nested Fractions

Nested fractions can seem intimidating, but breaking them down makes them much more understandable. These are fractions within fractions, which require careful handling.

• First, simplify the innermost fraction, making sure it’s in its simplest form.
• Then look at each level of the fraction, simplifying systematically from the inside out.
• Often, rewriting the fraction using common denominators will be necessary, as seen in the solution when dealing with \( \frac{a}{1+\frac{a}{a+1}} \).
• Multiply by reciprocals to resolve the nested fraction into a simpler expression.

In our example, the process involved manipulating \(\frac{a}{\frac{2a+1}{a+1}}\) by using the reciprocal, ultimately simplifying the expression to make it easier to work with. This breaks down complex problems step by step, making them approachable.