Problem 110 - Elementary Algebra

Question

For the following problems, evaluate each numerical expression. $$ 28 \cdot 14^{-1} $$

Step-by-Step Solution

Answer

Answer: The evaluated value of the given numerical expression is 2.

Step 1: Find the reciprocal of 14

To find the reciprocal of a number, we just take the fraction of that number with 1 as the numerator. So the reciprocal of 14 will be: $$ 14^{-1} = \frac{1}{14} $$

Step 2: Multiply 28 by the reciprocal of 14

Now that we have the reciprocal of 14, we will multiply it by 28: $$ 28 \cdot 14^{-1} = 28 \cdot \frac{1}{14} $$

Step 3: Simplify the expression

Now, we need to simplify the expression by performing the multiplication: $$ 28 \cdot \frac{1}{14} = \frac{28}{14} $$ Since 28 and 14 have a common factor of 14, we can simplify the fraction by dividing both the numerator and the denominator by 14: $$ \frac{28}{14} = \frac{28\div14}{14\div14} = \frac{2}{1} $$ Therefore, the evaluated value of the given numerical expression is: $$ 28 \cdot 14^{-1} = 2 $$

Key Concepts

Multiplying Fractions

Multiplying fractions may seem tricky at first, but with a little practice, it becomes straightforward. When you multiply fractions, you need to follow these simple steps:

• Multiply the numerators (the top numbers of the fractions) together to get the new numerator.
• Multiply the denominators (the bottom numbers of the fractions) together to get the new denominator.

In our example, you started with the expression \[ 28 \cdot \frac{1}{14} \]Here's why it worked:

• 28 can be written as \( \frac{28}{1} \) as a fraction. This means the multiplication becomes \( \frac{28}{1} \cdot \frac{1}{14} \).
• Multiplying the numerators: \( 28 \cdot 1 = 28 \).
• Multiplying the denominators: \( 1 \cdot 14 = 14 \).
• Thus, \( \frac{28}{1} \cdot \frac{1}{14} \) equals \( \frac{28}{14} \).

Multiplying fractions can often lead to larger numbers, which is why it helps to simplify them. This takes us to our next point.

Simplifying Fractions

Simplifying fractions means making them as simple as possible. This makes it easier to work with them. A fraction is simplified when both the numerator and the denominator have no common factors other than 1. Here's how you do it:

• Identify the greatest common divisor (GCD) of both numbers.
• Divide both the numerator and the denominator by the GCD.

Let's look at our earlier result:\[ \frac{28}{14} \]Both 28 and 14 have a common factor of 14:

• Divide the numerator (28) by 14, which gives 2.
• Divide the denominator (14) by 14, which gives 1.

Therefore, \( \frac{28}{14} \) simplifies to \( \frac{2}{1} \), or simply 2. Always check if your fraction can be simplified further. In this case, it couldn't go any simpler than 2. Simplifying not only makes expressions easier to handle but often provides insight into the relationships between numbers.

Numerical Expressions

A numerical expression is a mathematical phrase involving numbers and operation symbols but no equal sign. Evaluating numerical expressions involves carrying out operations to arrive at a single number. Take our expression \( 28 \cdot 14^{-1} \) as an example:

• First, understand what each part of the expression means. Here, \( 14^{-1} \) indicates multiplying by the reciprocal of 14.
• Convert \( 14^{-1} \) to \( \frac{1}{14} \).
• Then, perform the multiplication \( 28 \cdot \frac{1}{14} \) as shown in the previous sections.

By performing these steps, we simplified and evaluated the expression to find that it equals 2. Numerical expressions require careful processing of their components to arrive at a result. Once you get the hang of it, they become much easier to manage, allowing for smooth calculations and problem-solving.