Problem 105 - Intermediate Algebra

Question

Simplify each expression. $$ (16 b+8)\left(\frac{5}{4}\right)-8 b $$

Step-by-Step Solution

Answer

The simplified expression is \(12b + 10\).

Step 1: Distribute the Fraction

Apply the distributive property to the expression \((16b + 8)\left(\frac{5}{4}\right)\). Distribute \(\frac{5}{4}\) to each term in the parenthesis: \[ 16b \times \frac{5}{4} + 8 \times \frac{5}{4}. \] Simplifying these, we get: \[ \frac{80b}{4} + \frac{40}{4}. \] Thus, the expression becomes:\[ 20b + 10. \]

Step 2: Combine Like Terms

Now subtract the term \(8b\) from the expression obtained in Step 1: \[ 20b + 10 - 8b. \] Combine the like terms \(20b\) and \(-8b\):\[ (20b - 8b) + 10. \] This simplifies to: \[ 12b + 10. \]

Step 3: Final Simplified Expression

The expression is now simplified as much as possible. Therefore, the final simplified expression is: \[ 12b + 10 \].

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra. It allows us to simplify expressions by eliminating parentheses. Put simply, it means that you multiply each term inside the parentheses by the term outside of them. For instance, consider the expression \((16b + 8)\left(\frac{5}{4}\right)\).
To apply the distributive property here, you must multiply both \(16b\) and \(8\) by \(\frac{5}{4}\):

• \(16b \times \frac{5}{4}\)
• \(8 \times \frac{5}{4}\)

After applying these multiplications, we get \(\frac{80b}{4}\) and \(\frac{40}{4}\) respectively. These further simplify to \(20b\) and \(10\). By utilizing the distributive property, the expression condenses into \(20b + 10\). This process helps make complex expressions more manageable, paving the path for further simplification.

Combining Like Terms

Combining like terms is crucial to simplifying algebraic expressions. It involves combining terms with the same variable raised to the same power. Let's take a look at the step following the simplification of \(20b + 10\). There, we subtract \(8b\) from \(20b + 10\).
When you encounter like terms such as \(20b\) and \(-8b\), they can be combined:

• You subtract \(8b\) from \(20b\).
• This gives you \(12b\).

Now, if you also had a constant, like \(10\), sitting separately, you simply attach it to the result of the combined terms. Hence, \(20b + 10 - 8b\) simplifies to \(12b + 10\).
Combining like terms can significantly simplify expressions and make solving or understanding them much easier. Always remember to identify terms with similar characteristics to streamline your expressions effectively.

Fraction Multiplication

Understanding how to multiply fractions is essential when dealing with simplified expressions, especially in algebra. In our example, we had terms inside the parentheses that needed to be multiplied by the fraction \(\frac{5}{4}\). To handle fraction multiplication correctly, follow these steps:

• Multiply the denominator of the fraction by the constant coefficient of each variable, as shown in \(16b \times \frac{5}{4}\) and \(8 \times \frac{5}{4}\).
• Simplify the resulting fraction: \(\frac{80b}{4}\) becomes \(20b\), and \(\frac{40}{4}\) simplifies to \(10\).

When multiplying fractions:

• Consider multiplying the numerators together and the denominators together separately, then simplify the products if possible.
• This helps a lot in reducing the fraction to the lowest terms, thereby making your answer neat and concise.

Effectively managing fraction multiplication allows you to simplify complex expressions and work through mathematical problems more efficiently.