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 .

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 three decimal places.

 has two decimal places.

Our final answer will have two decimal places.

Solve the expression and round:

Rounding to two decimal places 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 eight significant figures.

 has six significant figures.

Our final answer will have six significant figures.

Solve the expression and round:

Rounding to six significant figures gives us .

This question requires that we first add, then multiply.

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.

7.1g has only one decimal place, so we round our answer to 20.7g.

Next, multiply the mass by the cost per gram. 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.

20.7g has only three significant figures, so we round our answer to $767.00. Note that the zeroes after the decimal are insignificant.

When performing addition or subtraction in chemistry you can only be as precise as the number with the fewest number of decimal not significant figures. Because  only has 1 decimal place the answer of the addition problem can only have 1 decimal place. All of the other numbers have more than 1 decimal place so that is why the other answers are incorrect.

When performing multiplication and division in chemistry, the answer to the question can only have as many significant figures as the number with the fewest number of significant figures.

 has 4 significant figures

 has 2 significant figures

 has 3 significant figures

 has 3 significant figures

Thus the answer must have 2 significant figures because  only has 2 significant figures.

disregarding significant figures. However to round to 2 significant figures, the answer becomes 

To round a number to 3 significant figures count to the 3rd significant figure and apply the rules of rounding to the next number. If the next number is 5 or higher round up, if it is 4 or below round down.

The 3rd significant figure is the 7. The next number is an 8 so the 7 is rounded up to an 8. As there are 4 places before the decimal point this answer must have 4 places

 rounds to  with three significant figures.

The other answers are wrong because:

 has 4 significant figures due to the decimal point

 rounds in the wrong direction

 only has 2 significant figures

 does not have the same amount of places before the decimal point

 follows the rules of significant figures.

1. Any digit 1-9 is always significant (So the 8 and the 3 are significant)

2. Any zero sandwiched between two digits 1-9 is always significant (No 0 in this case is significant because of this rule)

3. If the number is larger than 1 and there is no decimal point,  a zero at the end is NOT significant (trailing zero rule, no zeros are a part of this rule)

4. If the number is less than 1 any 0 before the first nonzero digit is not significant (leading zero rule)

In the number 0.000830 the bolded zeros are not significant because they are leading zeros

5. If a number has a decimal point, any zero that follows a non-zero digit is significant. 

0.000830 is the number. The last 0 follows a 3 and this number has a decimal point so it is significant. Totaling the significant figures, we have 3 sig figs.

0.000830  3 Significant Figures.