Problem 22 - Fundamentals of Mathematics

Question

Use a calculator to find each value. $$106^{3}-17^{4}$$

Step-by-Step Solution

Answer

The result is 1,107,495.

Step 1: Calculate 106 Cubed

First, use a calculator to find the value of \(106^3\). This means multiplying 106 by itself three times: \(106 \times 106 \times 106\). Upon calculation, \(106^3 = 1,191,016\).

Step 2: Calculate 17 to the Power of 4

Now, calculate \(17^4\). Multiply 17 by itself four times: \(17 \times 17 \times 17 \times 17\). Using a calculator, \(17^4 = 83,521\).

Step 3: Subtract 17^4 from 106^3

Subtract the result of \(17^4\) from \(106^3\). This involves doing the subtraction: \(1,191,016 - 83,521\). The result of the subtraction is \(1,107,495\).

Key Concepts

Exponentiation

Exponentiation is a fundamental arithmetic operation that extends the concept of multiplication. When you see an expression like \(106^3\), this refers to 106 raised to the power of 3, also known as "106 cubed". It means multiplying the number 106 by itself a total of three times:

• First, multiply 106 by 106.
• Then, take that result and multiply it by 106 again.

Calculations involving exponentiation often require the use of calculators, especially when dealing with large numbers or higher powers, to prevent errors and save time. In our specific example, calculating \(106^3\) resulted in 1,191,016, showcasing the rapid growth in size as numbers are repeatedly multiplied.

Step-by-step problem solving

Breaking down mathematical problems into smaller, more manageable steps can make complex operations much simpler. This approach was used in the exercise, where each part of the problem was solved one by one.

• First, the problem was divided into two separate calculations: \(106^3\) and \(17^4\).
• Second, each calculation was done individually using a calculator.
• Finally, the results from these calculations were used in a subtraction operation.

This step-by-step technique reduces errors and ensures that each part of the problem is correctly understood and solved. By isolating each section of the problem, students learn to focus on one operation at a time, resulting in a clearer understanding of the overall process.

Mathematical subtraction

Subtraction is a basic arithmetic operation that involves finding the difference between two numbers. In the context of the problem \(106^3 - 17^4\), the subtraction occurs after both numbers have been fully calculated.
In our case, it involves subtracting 83,521 (the result of \(17^4\)) from 1,191,016 (the result of \(106^3\)). Using a calculator can help make subtraction of large numbers trivial, allowing for speed and accuracy.
When performing subtraction on a calculator:

• Ensure digits are accurately entered to avoid mistakes.
• Double-check the calculation if the numbers are particularly close in value or if there is potential for negative results.

The resulting value of 1,107,495 in our operation showcases subtraction's role in determining differences between large computed numbers.

Arithmetical operations

Arithmetical operations include basic mathematical processes like addition, subtraction, multiplication, and division. These operations build the foundation of all mathematical problem-solving. In the exercise, these operations are represented by exponentiation and subtraction.

• **Exponentiation** involves multiplying numbers by themselves a number of times, representing repeated multiplication.
• **Subtraction** finds differences between numbers, important for comparing and deriving new values.

The combination of these operations allows students to solve complex expressions efficiently. The use of a calculator to handle these operations is crucial, particularly as the complexity of numbers increases. Utilizing technological tools helps ensure accuracy and reinforces numerical understanding through repeated practices.