Problem 66 - Elementary Algebra

Question

Write the following problems using scientific notation. $$ 46,000,000,000,000,000 $$

Step-by-Step Solution

Answer

Question: Convert the number $$46,000,000,000,000,000$$ into scientific notation. Answer: $$4.6 \times 10^{16}$$.

Step 1: Determine the significant figures

First, identify the significant figures in the given number. In this case, the significant figures are 46.

Step 2: Rewrite the number between 1 and 10

To rewrite the significant figures as a number between 1 and 10, we need to move the decimal point to the left, placing it between the two significant figures. So, the number becomes $$4.6$$.

Step 3: Count the number of decimal places moved

Next, we count how many decimal places the decimal point was moved to the left. In this case, the decimal point was moved 16 places to the left.

Step 4: Write the number as the product of the first number and a power of 10

Now, we express the given number as a product of the number between 1 and 10 obtained in step 2 and a power of 10, the exponent of which is the number of decimal places moved in step 3. Therefore, the scientific notation of the given number is $$4.6 \times 10^{16}$$.

Key Concepts

Significant Figures

Significant figures represent the digits in a number that carry meaningful contributions to its precision. In the significant figures of a number, every non-zero digit is always significant. Leading zeros are not counted, but any zeros between significant digits or at the end after the decimal point are included.
For example, in the number 46,000,000,000,000,000, only the digits '4' and '6' are considered significant figures because they are the non-zero figures representing the true value of this huge number. Recognizing these figures allows us to accurately express the number without losing precision when converting to scientific notation.

Decimal Point

The decimal point is a crucial component when converting a number to scientific notation. It indicates where the whole number ends and the fractional part begins. To express a number in scientific notation, we usually adjust where the decimal point is until we have a single non-zero digit to its left.
In the process of changing 46,000,000,000,000,000 into scientific notation, the decimal point originally at the end of the number is shifted leftwards. It is moved so that only the significant figures '4.6' are immediately followed by the decimal point. By positioning the decimal point between '4' and '6', we will express the number as a value between 1 and 10, a key format in scientific notation.

Power of 10

A power of 10 is used in scientific notation to express large or small numbers efficiently. It's the mathematical operation where 10 is multiplied by itself a certain number of times, indicated by the exponent. The concept helps in compressing and managing numbers that may have a very large amount of digits.
For instance, when we convert the number 46,000,000,000,000,000, we move the decimal point 16 places to the left to form the number 4.6. This action is then mathematically balanced by multiplying by 10 raised to the 16th power, which reflects every instance the decimal was moved. Hence, \[4.6 \times 10^{16}\] represents the original number in a more convenient form.