Problem 84 - Elementary Algebra

Question

Find the value of each of the following expressions. \(F=\left(p_{1}-p_{2}\right) r^{4} \cdot 9 . \quad\) Find \(F\) if \(p_{1}=12, p_{2}=7, r=2 .\)

Step-by-Step Solution

Answer

Answer: The value of F is 720.

Step 1: Identify Given Values

The given values are: \(p_{1} = 12, p_{2} = 7,\) and \(r = 2\).

Step 2: Calculate the Difference of \(p_{1}\) and \(p_{2}\)

Subtract \(p_{2}\) from \(p_{1}\): \(p_{1} - p_{2} = 12 - 7 = 5\).

Step 3: Calculate the Fourth Power of \(r\)

Raise \(r\) to the power of 4: \(r^4 = 2^4 = 16\).

Step 4: Multiply the Difference of \(p_{1}\) and \(p_{2}\) by \(r^4\)

Multiply \((p_{1} - p_{2})\) by \(r^4\): \((5)(16) = 80\).

Step 5: Multiply the Result by 9

Finally, multiply the result by 9: \(80 \cdot 9 = 720\).

Step 6: Write the Final Solution

The value of \(F\) is 720.

Key Concepts

Exponents

Exponents are an important part of algebraic expressions as they represent repeated multiplication of a number by itself. In this exercise, we have the expression \( r^4 \), which indicates that the number \( r \) (where \( r = 2 \)) is multiplied by itself four times. This means we calculate it as \( 2 \times 2 \times 2 \times 2 \). Let's break this down further:

• First, multiply \( 2 \times 2 \) to get 4.
• Next, multiply the result by 2, so \( 4 \times 2 = 8 \).
• Finally, multiply 8 by 2, resulting in 16.

Thus, \( r^4 = 16 \). Exponents can make expressions grow large quickly, hence understanding how to handle them is vital in solving algebraic expressions. By understanding the basic process of how exponents work, you're one step closer to mastering algebra.

Subtraction

Subtraction is a basic arithmetic operation that involves finding the difference between two numbers. It is a fundamental step in many algebraic expressions. In our exercise, we start with \( p_1 \ - p_2 \). This tells us to subtract \( p_2 \) from \( p_1 \):

• The given values are \( p_1 = 12 \) and \( p_2 = 7 \).
• Perform the subtraction: \( 12 - 7 \).
• We find that \( 12 - 7 = 5 \).

This operation reduces the expression to a simpler form that can be easily used in subsequent steps. Subtraction allows us to manage and manipulate numbers effectively, serving as a building block for more complex calculations.

Multiplication

Multiplication helps combine quantities and is essential for calculating the final result in expressions. In this problem, we need to multiply a few different values together. The first multiplication involves the result of the subtraction, which is \( 5 \), and the exponentiated term \( r^4 = 16 \):

• Multiply the difference \( 5 \) by \( 16 \) to get \( 5 \times 16 = 80 \).

After completing this step, another multiplication is needed to reach the answer desired in the exercise:

• We multiply the product \( 80 \) by \( 9 \): \( 80 \times 9 = 720 \).

Multiplication not only simplifies combinations of numbers but also efficiently scales the results, showing why this fundamental operation is pivotal in algebra.

Order of Operations

The order of operations is a set of rules that ensure calculations are performed in a consistent manner. Following this order is crucial for reaching the correct answer in any algebraic expression. In this problem, to find the value of \( F \), we must carefully follow these steps:

• Parentheses: Resolve any calculations inside parentheses first. Here \( p_1 - p_2 = 5 \).
• Exponents: Next, handle any exponents. In our case, calculate \( r^4 = 16 \).
• Multiplication: Perform multiplication or division from left to right. Multiply \( 5 \) by \( 16 \) to get \( 80 \), then multiply the result by \( 9 \) to arrive at \( 720 \).
• Addition or Subtraction: Lastly, handle any addition or subtraction. In our example, this step is already done since subtraction was resolved in the parentheses step.

By following the order of operations, we ensure that each calculation is done sequentially and correctly without any errors, providing the accurate result for the problem.