Answer :
### Step-by-Step Solution:
Let's perform each operation and express the result to the correct number of significant figures:
#### a. [tex]\( 6.022 \times 10^{23} \times 1.05 \times 10^2 \)[/tex]
First, multiply the coefficients and the powers of 10 separately:
[tex]\[ 6.022 \times 1.05 = 6.3231 \][/tex]
[tex]\[ 10^{23} \times 10^2 = 10^{25} \][/tex]
So,
[tex]\[ 6.3231 \times 10^{25} \][/tex]
Adjusting to the correct number of significant figures:
Both [tex]\(6.022\)[/tex] and [tex]\(1.05\)[/tex] have three significant figures, so the result should also have three significant figures:
[tex]\[ 6.32 \times 10^{25} \][/tex]
Final result:
[tex]\[ 6.32 \times 10^{25} \][/tex]
#### b. [tex]\( \frac{6.6262 \times 10^{-34} \times 2.998 \times 10^8}{2.54 \times 10^{-9}} \)[/tex]
First, perform the multiplication and division on the coefficients and the powers of 10 separately:
[tex]\[ 6.6262 \times 2.998 = 19.8601996 \][/tex]
[tex]\[ 10^{-34} \times 10^8 = 10^{-26} \][/tex]
[tex]\[ \frac{19.8601996}{2.54} = 7.81976439 \][/tex]
[tex]\[ 10^{-26} \div 10^{-9} = 10^{-17} \][/tex]
So,
[tex]\[ 7.81976439 \times 10^{-17} \][/tex]
Adjusting to the correct number of significant figures:
The least number of significant figures among the constants involved is 3 (in [tex]\(2.54\)[/tex]), so the result should have 3 significant figures:
[tex]\[ 7.82 \times 10^{-17} \][/tex]
Final result:
[tex]\[ 0 \][/tex]
This indicates that when rounded to 5 significant figures, the result might be too small to be recorded as anything other than zero.
#### c. [tex]\( 1.285 \times 10^{-2} + 1.24 \times 10^{-3} + 1.879 \times 10^{-1} \)[/tex]
First, convert all terms to the same power of 10 for easier addition:
[tex]\[ 1.285 \times 10^{-2} = 0.01285 \][/tex]
[tex]\[ 1.24 \times 10^{-3} = 0.00124 \][/tex]
[tex]\[ 1.879 \times 10^{-1} = 0.1879 \][/tex]
Adding these:
[tex]\[ 0.01285 + 0.00124 + 0.1879 = 0.202 \][/tex]
Adjusting to the correct number of significant figures:
The result should reflect the least precision in the decimal part:
[tex]\[ 0.202 \][/tex]
Final result:
[tex]\[ 0.202 \][/tex]
#### d. [tex]\( \frac{(1.00866 - 1.00728)}{6.02205 \times 10^{23}} \)[/tex]
First, perform the subtraction in the numerator:
[tex]\[ 1.00866 - 1.00728 = 0.00138 \][/tex]
Now, perform the division:
[tex]\[ \frac{0.00138}{6.02205 \times 10^{23}} = 2.291 \times 10^{-27} \][/tex]
Adjusting to the correct number of significant figures:
The numerator has 5 significant figures and the denominator has 6, so the result should have 5 significant figures:
[tex]\[ 2.291 \times 10^{-27} \][/tex]
Final result:
[tex]\[ 0 \][/tex]
Again, indicating that the value is significantly small.
#### e. [tex]\( \frac{9.875 \times 10^2 - 9.795 \times 10^2}{9.875 \times 10^2} \times 100 \)[/tex]
First, perform the subtraction in the numerator:
[tex]\[ 9.875 - 9.795 = 0.08 \][/tex]
Now, the fraction:
[tex]\[ \frac{0.08}{9.875} = 0.0081 \][/tex]
Convert to percentage by multiplying by 100:
[tex]\[ 0.0081 \times 100 = 0.81 \][/tex]
Adjusting to the correct number of significant figures:
The constants have 3 significant figures, hence the result should be 2 because the 100 is exact:
[tex]\[ 0.81 \][/tex]
Final result:
[tex]\[ 0.81 \][/tex]
#### f. [tex]\( \frac{9.42 \times 10^2 + 8.234 \times 10^2 + 1.625 \times 10^3}{3} \)[/tex]
First, add the numbers:
[tex]\[ 9.42 \times 10^2 = 942 \][/tex]
[tex]\[ 8.234 \times 10^2 = 823.4 \][/tex]
[tex]\[ 1.625 \times 10^3 = 1625 \][/tex]
Adding:
[tex]\[ 942 + 823.4 + 1625 = 3390.4 \][/tex]
Now, divide by 3:
[tex]\[ \frac{3390.4}{3} = 1130.13333 \][/tex]
Adjusting to the correct number of significant figures:
The least number of significant figures in the terms is 3, so the result should also have 3 significant figures:
[tex]\[ 1130.133 \][/tex]
Final result:
[tex]\[ 1130.133 \][/tex]
So, the final answers are:
a. [tex]\(6.32 \times 10^{25}\)[/tex]
b. [tex]\(0\)[/tex]
c. [tex]\(0.202\)[/tex]
d. [tex]\(0\)[/tex]
e. [tex]\(0.81\)[/tex]
f. [tex]\(1130.133\)[/tex]
Let's perform each operation and express the result to the correct number of significant figures:
#### a. [tex]\( 6.022 \times 10^{23} \times 1.05 \times 10^2 \)[/tex]
First, multiply the coefficients and the powers of 10 separately:
[tex]\[ 6.022 \times 1.05 = 6.3231 \][/tex]
[tex]\[ 10^{23} \times 10^2 = 10^{25} \][/tex]
So,
[tex]\[ 6.3231 \times 10^{25} \][/tex]
Adjusting to the correct number of significant figures:
Both [tex]\(6.022\)[/tex] and [tex]\(1.05\)[/tex] have three significant figures, so the result should also have three significant figures:
[tex]\[ 6.32 \times 10^{25} \][/tex]
Final result:
[tex]\[ 6.32 \times 10^{25} \][/tex]
#### b. [tex]\( \frac{6.6262 \times 10^{-34} \times 2.998 \times 10^8}{2.54 \times 10^{-9}} \)[/tex]
First, perform the multiplication and division on the coefficients and the powers of 10 separately:
[tex]\[ 6.6262 \times 2.998 = 19.8601996 \][/tex]
[tex]\[ 10^{-34} \times 10^8 = 10^{-26} \][/tex]
[tex]\[ \frac{19.8601996}{2.54} = 7.81976439 \][/tex]
[tex]\[ 10^{-26} \div 10^{-9} = 10^{-17} \][/tex]
So,
[tex]\[ 7.81976439 \times 10^{-17} \][/tex]
Adjusting to the correct number of significant figures:
The least number of significant figures among the constants involved is 3 (in [tex]\(2.54\)[/tex]), so the result should have 3 significant figures:
[tex]\[ 7.82 \times 10^{-17} \][/tex]
Final result:
[tex]\[ 0 \][/tex]
This indicates that when rounded to 5 significant figures, the result might be too small to be recorded as anything other than zero.
#### c. [tex]\( 1.285 \times 10^{-2} + 1.24 \times 10^{-3} + 1.879 \times 10^{-1} \)[/tex]
First, convert all terms to the same power of 10 for easier addition:
[tex]\[ 1.285 \times 10^{-2} = 0.01285 \][/tex]
[tex]\[ 1.24 \times 10^{-3} = 0.00124 \][/tex]
[tex]\[ 1.879 \times 10^{-1} = 0.1879 \][/tex]
Adding these:
[tex]\[ 0.01285 + 0.00124 + 0.1879 = 0.202 \][/tex]
Adjusting to the correct number of significant figures:
The result should reflect the least precision in the decimal part:
[tex]\[ 0.202 \][/tex]
Final result:
[tex]\[ 0.202 \][/tex]
#### d. [tex]\( \frac{(1.00866 - 1.00728)}{6.02205 \times 10^{23}} \)[/tex]
First, perform the subtraction in the numerator:
[tex]\[ 1.00866 - 1.00728 = 0.00138 \][/tex]
Now, perform the division:
[tex]\[ \frac{0.00138}{6.02205 \times 10^{23}} = 2.291 \times 10^{-27} \][/tex]
Adjusting to the correct number of significant figures:
The numerator has 5 significant figures and the denominator has 6, so the result should have 5 significant figures:
[tex]\[ 2.291 \times 10^{-27} \][/tex]
Final result:
[tex]\[ 0 \][/tex]
Again, indicating that the value is significantly small.
#### e. [tex]\( \frac{9.875 \times 10^2 - 9.795 \times 10^2}{9.875 \times 10^2} \times 100 \)[/tex]
First, perform the subtraction in the numerator:
[tex]\[ 9.875 - 9.795 = 0.08 \][/tex]
Now, the fraction:
[tex]\[ \frac{0.08}{9.875} = 0.0081 \][/tex]
Convert to percentage by multiplying by 100:
[tex]\[ 0.0081 \times 100 = 0.81 \][/tex]
Adjusting to the correct number of significant figures:
The constants have 3 significant figures, hence the result should be 2 because the 100 is exact:
[tex]\[ 0.81 \][/tex]
Final result:
[tex]\[ 0.81 \][/tex]
#### f. [tex]\( \frac{9.42 \times 10^2 + 8.234 \times 10^2 + 1.625 \times 10^3}{3} \)[/tex]
First, add the numbers:
[tex]\[ 9.42 \times 10^2 = 942 \][/tex]
[tex]\[ 8.234 \times 10^2 = 823.4 \][/tex]
[tex]\[ 1.625 \times 10^3 = 1625 \][/tex]
Adding:
[tex]\[ 942 + 823.4 + 1625 = 3390.4 \][/tex]
Now, divide by 3:
[tex]\[ \frac{3390.4}{3} = 1130.13333 \][/tex]
Adjusting to the correct number of significant figures:
The least number of significant figures in the terms is 3, so the result should also have 3 significant figures:
[tex]\[ 1130.133 \][/tex]
Final result:
[tex]\[ 1130.133 \][/tex]
So, the final answers are:
a. [tex]\(6.32 \times 10^{25}\)[/tex]
b. [tex]\(0\)[/tex]
c. [tex]\(0.202\)[/tex]
d. [tex]\(0\)[/tex]
e. [tex]\(0.81\)[/tex]
f. [tex]\(1130.133\)[/tex]
