Problem 75 - Elementary Algebra

Question

\(-\frac{3}{5} a\left(10 a+\frac{1}{2}\right)\)

Step-by-Step Solution

Answer

-6a^2 - \frac{3}{10} a

Step 1: Expand the Expression

Distribute \( -\frac{3}{5} a \) through the terms inside the parenthesis: \[ -\frac{3}{5} a \left( 10a + \frac{1}{2} \right). \]

Step 2: Distribute to the First Term

Multiply \( -\frac{3}{5} a \) by \( 10a \): \[ -\frac{3}{5} a \cdot 10a = -\frac{3}{5} \cdot 10 \cdot a^2 = -6a^2 \]

Step 3: Distribute to the Second Term

Multiply \( -\frac{3}{5} a \) by \( \frac{1}{2} \): \[ -\frac{3}{5} a \cdot \frac{1}{2} = -\frac{3}{5} \cdot \frac{1}{2} \cdot a = -\frac{3}{10} a \]

Step 4: Combine the Results

Combine the results from Step 2 and Step 3: \[ -6a^2 - \frac{3}{10} a. \]

Key Concepts

Distribution Property

The Distribution Property is a useful tool in algebra to simplify expressions. It allows you to multiply each term inside the parenthesis by an external factor. This property is especially important when dealing with algebraic expressions containing variables.
Here's a simple rule to remember: distribute the factor outside the parenthesis to every term inside.

• For the exercise given, we distribute \(-\frac{3}{5} a\) to both \(10a\) and \(\frac{1}{2}\).
• This involves multiplying \( -\frac{3}{5} a \) by \(10a\) first.
• Next, we multiply \( -\frac{3}{5} a \) by \( \frac{1}{2}\).

This systematic approach ensures every term inside the parenthesis is appropriately multiplied and contributes to simplifying the expression accurately.

Combining Like Terms

Combining like terms is another crucial concept in algebra. It involves adding or subtracting terms with the same variable and the same exponent.
In the exercise, after distributing, you get two terms: \(-6a^2\) and \(-\frac{3}{10}a\).

• Check if the terms have the same variable raised to the same power.
• In our example, the terms have different powers and cannot be combined.

Remember, only terms with like variables and exponents can be combined. This simplifies the expression, making it easier to work with in further calculations.

Simplifying Expressions

Simplifying expressions involves reducing them to their most concise form. This process can include distributing factors, combining like terms, and reducing fractions.
Let's tie everything back to the original exercise:

• First, we distributed the \(-\frac{3}{5} a\) to each term inside the parenthesis.
• Next, we performed the multiplications to get \(-6a^2 \) and \(-\frac{3}{10}a\).

The final step is checking if further simplifications are possible. In this case, the expression \(-6a^2 - \frac{3}{10}a\) is already in its simplest form.
This three-step method ensures your expressions are as simplified as possible, easing your path in solving algebraic equations.