Problem 59 - Elementary Algebra

Question

Write the following problems using scientific notation. $$ 746,000 $$

Step-by-Step Solution

Answer

Question: Convert the number 746,000 into scientific notation. Answer: The number 746,000 in scientific notation is $$7.46 \times 10^5$$.

Step 1: Identify the coefficient

To find the coefficient, we need to rewrite the given number so that it is between 1 and 9.99 (inclusive). To do this, we should move the decimal point to the left, such that we obtain a number within the specified range. In this case, the decimal point must be moved 5 places to the left, resulting in the number 7.46.

Step 2: Identify the exponent

Next, we need to determine the exponent of 10. The exponent should be equal to the number of places we moved the decimal point in Step 1. In this case, we moved the decimal point 5 places to the left, so the exponent is 5.

Step 3: Write the number in scientific notation

Finally, we can combine the coefficient from Step 1 (7.46) and the exponent from Step 2 (5) to write the given number in scientific notation. The final result in scientific notation is: $$ 7.46 \times 10^5 $$

Key Concepts

Understanding Coefficient in Scientific Notation

The coefficient is a critical component of scientific notation and is essential for simplification and comprehension. In scientific notation, the coefficient is the part of the number that remains after adjusting it to fall between 1 and 9.99.

To determine the coefficient, we often need to move the decimal point to create a manageable number. For example, if you start with 746,000, you move the decimal point so that the number is within the range of 1 to 9.99. This means shifting the decimal 5 places to the left, yielding 7.46.

In scientific terms, the coefficient makes it easier for others to quickly understand the scale of the number. It succinctly represents a large or small number without listing all its digits, which can be cumbersome.

• The coefficient must always be more than or equal to 1.
• It must be less than 10.
• Any number, whether large or small, eventually becomes easy to read and write with this simplification.

Decoding the Exponent

In scientific notation, the exponent tells you how many times to multiply the coefficient by 10. It's derived from how many places the decimal point is moved from its original location.

When converting a number like 746,000, the decimal point is moved 5 places left to get the coefficient 7.46; thus, the exponent becomes 5. The expression becomes 7.46 × 10^5.

This exponent greatly aids in understanding the scale of a number—whether it’s very large or very small. It uses powers of ten, which are intuitive for multiplication and division, making calculations simpler and quicker.

• The exponent is positive if you move the decimal point left, which indicates a large number.
• Conversely, moving the decimal point to the right results in a negative exponent, indicating a small number.
• This compact format is ideal for expressing sizes ranging from the microscopic level to astronomical scales.

Decimal Point Placement in Scientific Notation

The placement of the decimal point is pivotal when converting a number to scientific notation. It precisely dictates the values of the coefficient and the exponent.

If you are converting a number like 746,000 to scientific notation, the decimal point initially sits at the end (although it's not written). To convert it, you need to make it sit after the number 7, resulting in 7.46. This shift of 5 places is what sets the exponent.

The goal is to ensure that the coefficient remains between 1 and 9.99. The proper placement of the decimal point determines this, making the whole notation possible and purposeful.

• The correct movement allows large numbers to shrink and tiny numbers to be more pronounced, allowing for easier reading.
• Ensuring the decimal is relocated correctly avoids misrepresenting the number's true value.
• Understanding decimal placement helps in both computational calculations and theoretical applications efficiently.