Problem 353 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ 15 \cdot \frac{3}{5}(4 d+10) $$

Step-by-Step Solution

Answer

The simplified expression is \( 36d + 90 \).

Step 1 - Apply the Distributive Property

Use the Distributive Property to distribute the factor outside the parentheses to each term inside the parentheses. So, distribute \(\frac{3}{5}\) to both \(4d\) and \(10\). This gives: \[ \frac{3}{5} \times (4d) + \frac{3}{5} \times 10 \]

Step 2 - Simplify Each Term

Simplify each term separately. \[ \frac{3}{5} \times 4d = \frac{12d}{5} \] and \[ \frac{3}{5} \times 10 = 6 \]. So the expression now is \[ 15 \times \frac{12d}{5} + 15 \times 6 \]

Step 3 - Simplify Further

Simplify further by multiplying. For \( 15 \times \frac{12d}{5} \), multiply the numerators and denominators: \[ 15 \times \frac{12d}{5} = \frac{15 \times 12d}{5} = \frac{180d}{5} = 36d \]. Similarly, for the constant term, \( 15 \times 6 = 90 \)

Step 4 - Combine the Results

Combine the simplified results: \[ 36d + 90 \]

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to their simplest form. This makes it easier to work with and understand them. The basic goal is to perform arithmetic operations to eliminate any unnecessary terms or factors.
For example, simplifying an expression might include:

• Combining like terms
• Using the distributive property

In the given exercise, we start with a complex expression: \(15 \times \frac{3}{5}(4d + 10)\). We first need to apply the distributive property to break down the expression into simpler parts.
After distributing, we simplify each part by performing the multiplication. By following the steps carefully, we can transform the original expression into its simplest form.

Distributive Property

The distributive property is a fundamental concept in algebra. It allows us to multiply a single term by each term inside a parenthesis. The distributive property states:
\(a(b + c) = ab + ac\)
This property is useful because it helps break down more complicated expressions into simpler components.
In our exercise, we use the distributive property to distribute \(\frac{3}{5}\) to both \(4d\) and \(10\). This gives us:
\(\frac{3}{5} \times 4d + \frac{3}{5} \times 10\)
After distributing, we proceed to simplify each term individually.

Combining Like Terms

Combining like terms is another essential technique in algebra. It involves merging terms that have the same variables and exponents.
For instance, \(3d + 5d = 8d\) because both terms have the same variable \(d\). However, \(3d + 5x\) cannot be combined because the variables are different.
In our simplified expression \(15 \cdot \frac{12d}{5} + 15 \times 6\), we further simplify it to obtain:
\(36d + 90\)
By combining like terms correctly, we ensure that the final expression is in its simplest form, making it much easier to work with in subsequent steps.