Problem 6 - Fundamentals of Mathematics

Question

Write each number without exponents. $$1,739^{2}$$

Step-by-Step Solution

Answer

The number 1,739 without exponents is 3,024,121.

Step 1. Understand the Problem

We need to expand the number \(1,739^2\) by multiplying \(1,739\) by itself, as the exponent 2 indicates that the number is squared.

Step 2. Set Up the Multiplication

Write down the multiplication as \(1,739 \times 1,739\). This setup helps visualize the task as a long multiplication problem.

Step 3. Break Down the Multiplication

Use long multiplication to break down the problem: multiply \(1,739\) by each digit of \(1,739\), starting from the right-most digit.

Step 4. Multiply by the Ones Digit

Multiply \(1,739\) by 9 (the ones digit of 1,739):\[\begin{align*} 1,739 \times 9 &= 15,651 \end{align*}\]

Step 5. Multiply by the Tens Digit

Now multiply \(1,739\) by 3 (the tens digit) and add a zero because it’s in the ten’s place:\[\begin{align*} 1,739 \times 30 &= 52,170 \end{align*}\]

Step 6. Multiply by the Hundreds Digit

Multiply \(1,739\) by 7 (the hundreds digit) and add two zeros as it's in the hundred's place:\[\begin{align*} 1,739 \times 700 &= 1,217,300 \end{align*}\]

Step 7. Multiply by the Thousands Digit

Multiply \(1,739\) by 1 (the thousands digit) and add three zeros because it’s in the thousand’s place:\[\begin{align*} 1,739 \times 1,000 &= 1,739,000 \end{align*}\]

Step 8. Sum All the Products

Add the results of each step together to find the final result:\[\begin{align*} 15,651 \ +52,170 \ +1,217,300 \ +1,739,000 \ \hline 3,024,121 \end{align*}\]

Step 9. Conclusion

The result of expanding \(1,739^2\) is 3,024,121 after calculating each multiplication step and summing them correctly.

Key Concepts

Squaring Numbers

When we talk about squaring numbers, we mean multiplying a number by itself. This is a straightforward way to express how many times the number is multiplied. For example, the exercise asked to find the result of squaring 1,739. If we represent 1,739 squared with an exponent, it looks like this: - \(1,739^2\)
This tells us to multiply 1,739 by itself, resulting in a calculation you will later see laid out step by step.
There are some interesting properties of squaring numbers:

• Squaring a positive number always gives a positive result.
• Squaring zero results in zero.
• Even numbers squared remain even, while odd numbers squared remain odd.

By understanding these properties, you can anticipate the types of results you might encounter when squaring different numbers.

Long Multiplication

Long multiplication is a technique used to multiply larger numbers, making it manageable by breaking it down into smaller, sequential steps. In our example, we use long multiplication to expand \(1,739^2\).
Here's a glimpse of the process:

• First, we multiply the main number by each digit from right to left, taking note of their place value.
• We must add zeros or shift each product correctly to align with its place value (tens, hundreds, thousands, etc.).
• Each result is then stacked, ready to be summed together for the final answer.

This systematic approach is crucial in performing accurate calculations and is fundamental for problems like our example where smaller multiplications are combined into a single result.

Solving Multiplication Problems

Solving multiplication problems, especially larger ones, requires careful attention and a structured approach. In our example, solving \(1,739^2\) involved a series of multiplications which we carefully executed one at a time. Let's look at some essential tips for solving such problems:

• Break Down the Problem: Take the original number and decompose it based on its digits and their respective place values.
• Systematic Multiplication: Perform the multiplications starting from the smallest digit, ensuring that each result is properly aligned according to its place value.
• Accurate Addition: Finally, sum up all the aligned products carefully. This ensures no steps are missed, and the final result is precise.

The accuracy of a multiplication problem relies on your ability to maintain order and precision through each step, which is why following a standard method is highly recommended.