Scientific notation is used to simplify exceptionally complex numbers and to quickly present the number of significant figures in a given value. The value is converted to an exponent form using base ten, such that only a single-digit term with any given number of decimal places is used to represent the significant figures of the given value. Non-significant zeroes can be omitted from the leading term, and represented only in the base ten exponent.

The given number has only one significant figure (), so there will only be one digit multiplied by the base ten term.

To generate the single-digit leading term the decimal must be placed after the 4. Then count the digits to the left of the decimal (moving from right to left, these are  and ). Our ten will be raised to the power of negative two because there are two digits to the left of the final decimal placement.

Our final answer is .

Let's look at the rules of significant figures:

I. All nonzero digits are significant.

II. Any zero found anywhere between nonzero digits is significant.

III. Any zero to the right of the decimal point AND after a nonzero digit is significant.

IV. Any zero to the left of the decimal point AND after a nonzero digit is significant. Any zero found after a nonzero digit with no decimal point indication is NOT significant.

If we apply rule I, the answer should be two significant figures since  is a two-digit number. 

Let's look at the rules of significant figures:

I. All nonzero digits are significant.

II. Any zero found anywhere between nonzero digits is significant.

III. Any zero to the right of the decimal point AND after a nonzero digit is significant.

IV. Any zero to the left of the decimal point AND after a nonzero digit is significant. Any zero found after a nonzero digit with no decimal point indication is NOT significant.

Let's look at rule I first. In , there are two nonzero digits. According to rule II, the zero is significant as well.  has three significant figures. 

Let's look at the rules of significant figures:

I. All nonzero digits are significant.

II. Any zero found anywhere between nonzero digits is significant.

III. Any zero to the right of the decimal point AND after a nonzero digit is significant.

IV. Any zero to the left of the decimal point AND after a nonzero digit is significant. Any zero found after a nonzero digit with no decimal point indication is NOT significant.

Let's look at rule I.  only has one nonzero digit, and two zeroes. So far, there's one significant digit. There is no decimal point present, so according to rule IV the zeroes are insignificant.  only has one significant figure. 

Let's look at the rules of significant figures:

I. All nonzero digits are significant.

II. Any zero found anywhere between nonzero digits is significant.

III. Any zero to the right of the decimal point AND after a nonzero digit is significant.

IV. Any zero to the left of the decimal point AND after a nonzero digit is significant. Any zero found after a nonzero digit with no decimal point indication is NOT significant.

Let's look at rule I. In , there is only one nonzero digit and two zeroes present. That gives us one significant figure. Now, looking at rule III, the zero is to the right of the decimal but there is no nonzero digit to the left of it, making the leading zero insignificant. Finally, in rule IV, the zero after the decimal point is before the nonzero digit making and is also insignificant.

 contains only one significant figure.

Let's look at the rules of significant figures:

I. All nonzero digits are significant.

II. Any zero found anywhere between nonzero digits is significant.

III. Any zero to the right of the decimal point AND after a nonzero digit is significant.

IV. Any zero to the left of the decimal point AND after a nonzero digit is significant. Any zero found after a nonzero digit with no decimal point indication is NOT significant.

According to rule I, the two nonzero digits will be significant. The zeroes can be classified under rule II, as they all fall between two nonzero digits (the presence of the decimal here is negligible). The result is five sigificant figures in the given number, .

Let's look at the rules of significant figures:

I. All nonzero digits are significant.

II. Any zero found anywhere between nonzero digits is significant.

III. Any zero to the right of the decimal point AND after a nonzero digit is significant.

IV. Any zero to the left of the decimal point AND after a nonzero digit is significant. Any zero found after a nonzero digit with no decimal point indication is NOT significant.

Rule I gives us three significant figures, as there are three nonzero digits. Next, there are two zeroes positioned between two nonzero digits; these are both significant according to rule II . The zero to the right of the three obeys rule III and must be significant. All six of the digits in  are significant. 

Let's look at the rules of significant figures:

I. All nonzero digits are significant.

II. Any zero found anywhere between nonzero digits is significant.

III. Any zero to the right of the decimal point AND after a nonzero digit is significant.

IV. Any zero to the left of the decimal point AND after a nonzero digit is significant. Any zero found after a nonzero digit with no decimal point indication is NOT significant.

Rule I gives us only one nonzero significant figure. However, rule IV tells us that the zeroes are significant because there is a decimal present and the zeroes are following a nonzero digit. Therefore,  has four significant figures. 

When adding or subtracting, the final answer will have the same number of decimal places as the term with the fewest decimal places in the calculation. Always complete the calculation without rounding first to avoid errors.

 has one decimal place.

 has two decimal places.

Our final answer will have one decimal place.

Solve the expression and round:

Rounding to one decimal place gives us .

When multiplying or dividing, the final answer will have the same number of significant figures as the term with the fewest significant figures in the calculation. Always complete the calculation without rounding first to avoid errors.

 has three significant figures.

 has four significant figures.

Our final answer will have three significant figures.

Solve the expression and round:

Rounding to three significant figures gives us .