Answer :
Sure! Let's solve each of these operations step-by-step, ensuring that the results have the correct number of significant figures.
### a) [tex]\( 4.085 \times 10^{-3} \div 2.367 \)[/tex]
1. Calculate the division:
[tex]\[ 4.085 \times 10^{-3} \div 2.367 \approx 0.00172581326573722 \][/tex]
2. Determine the number of significant figures. The operand with the least significant figures is [tex]\(2.367\)[/tex] (4 significant figures).
3. Round the result to 4 significant figures:
[tex]\[ 0.001726 \][/tex]
Therefore, the result is [tex]\(0.001726\)[/tex] with 4 significant figures.
### b) [tex]\( 68 - 22.4 \)[/tex]
1. Calculate the subtraction:
[tex]\[ 68 - 22.4 = 45.6 \][/tex]
2. Determine the number of significant figures. The operand [tex]\(68\)[/tex] has 2 significant figures while [tex]\(22.4\)[/tex] has 3 significant figures. In subtraction or addition, the result should be rounded to the least number of decimal places, which is 0 decimal places (for [tex]\(68\)[/tex]).
3. Round the result:
[tex]\[ 46 \][/tex]
Therefore, the result is [tex]\(46\)[/tex] with 2 significant figures.
### c) [tex]\( \frac{6.780 \times 5.6}{345.18} \)[/tex]
1. Calculate the multiplication and division:
[tex]\[ \frac{6.780 \times 5.6}{345.18} \approx 0.10999478532939334 \][/tex]
2. Determine the number of significant figures. The operand [tex]\(5.6\)[/tex] has the least significant figures (2 significant figures).
3. Round the result to 2 significant figures:
[tex]\[ 0.11 \][/tex]
Therefore, the result is [tex]\(0.11\)[/tex] with 2 significant figures.
### d) [tex]\( 352.64 + 73.5 + 10 \)[/tex]
1. Calculate the addition:
[tex]\[ 352.64 + 73.5 + 10 = 436.14 \][/tex]
2. Determine the number of significant figures. The operand [tex]\(10\)[/tex] has 2 significant figures but no decimal places; hence, we consider the operand with no decimal places (10).
3. Round the result to 0 decimal places:
[tex]\[ 436 \][/tex]
Therefore, the result is [tex]\(436\)[/tex] with 0 decimal places (as significant figures).
### e) [tex]\( 3.64 \times 10^{-5} \times 4.5 \times 10^9 \)[/tex]
1. Calculate the multiplication:
[tex]\[ 3.64 \times 10^{-5} \times 4.5 \times 10^9 \approx 163800.0 \][/tex]
2. Determine the number of significant figures. The operands [tex]\(3.64 \times 10^{-5}\)[/tex] and [tex]\(4.5 \times 10^9\)[/tex] have the least significant number which is 2 significant figures.
3. Round the result to 2 significant figures:
[tex]\[ 1.6 \times 10^5 \][/tex]
Therefore, the result is [tex]\(1.6 \times 10^5\)[/tex] with 2 significant figures.
### f) [tex]\( \frac{42.47 - 2.690 + \log(3.110 \times 10^2)}{(12.30 - 2.804) \times 4.2001} \)[/tex]
1. Calculate each component:
[tex]\[ 42.47 - 2.690 + \log(3.110 \times 10^2) \approx 42.47 - 2.690 + 2.493 = 42.273 \][/tex]
[tex]\[ (12.30 - 2.804) \times 4.2001 \approx 9.496 \times 4.2001 \approx 39.877896 \][/tex]
2. Perform the final division:
[tex]\[ \frac{42.273}{39.877896} \approx 1.0598887230386589 \][/tex]
3. Determine the number of significant figures. The operand [tex]\(2.804\)[/tex] has the least significant figures (3 significant figures).
4. Round the result to 3 significant figures:
[tex]\[ 1.06 \][/tex]
Therefore, the result is [tex]\(1.06\)[/tex] with 3 significant figures.
In summary, the results with the correct number of significant figures are as follows:
a) [tex]\(0.001726\)[/tex]
b) [tex]\(46\)[/tex]
c) [tex]\(0.11\)[/tex]
d) [tex]\(436\)[/tex]
e) [tex]\(1.6 \times 10^5\)[/tex]
f) [tex]\(1.06\)[/tex]
### a) [tex]\( 4.085 \times 10^{-3} \div 2.367 \)[/tex]
1. Calculate the division:
[tex]\[ 4.085 \times 10^{-3} \div 2.367 \approx 0.00172581326573722 \][/tex]
2. Determine the number of significant figures. The operand with the least significant figures is [tex]\(2.367\)[/tex] (4 significant figures).
3. Round the result to 4 significant figures:
[tex]\[ 0.001726 \][/tex]
Therefore, the result is [tex]\(0.001726\)[/tex] with 4 significant figures.
### b) [tex]\( 68 - 22.4 \)[/tex]
1. Calculate the subtraction:
[tex]\[ 68 - 22.4 = 45.6 \][/tex]
2. Determine the number of significant figures. The operand [tex]\(68\)[/tex] has 2 significant figures while [tex]\(22.4\)[/tex] has 3 significant figures. In subtraction or addition, the result should be rounded to the least number of decimal places, which is 0 decimal places (for [tex]\(68\)[/tex]).
3. Round the result:
[tex]\[ 46 \][/tex]
Therefore, the result is [tex]\(46\)[/tex] with 2 significant figures.
### c) [tex]\( \frac{6.780 \times 5.6}{345.18} \)[/tex]
1. Calculate the multiplication and division:
[tex]\[ \frac{6.780 \times 5.6}{345.18} \approx 0.10999478532939334 \][/tex]
2. Determine the number of significant figures. The operand [tex]\(5.6\)[/tex] has the least significant figures (2 significant figures).
3. Round the result to 2 significant figures:
[tex]\[ 0.11 \][/tex]
Therefore, the result is [tex]\(0.11\)[/tex] with 2 significant figures.
### d) [tex]\( 352.64 + 73.5 + 10 \)[/tex]
1. Calculate the addition:
[tex]\[ 352.64 + 73.5 + 10 = 436.14 \][/tex]
2. Determine the number of significant figures. The operand [tex]\(10\)[/tex] has 2 significant figures but no decimal places; hence, we consider the operand with no decimal places (10).
3. Round the result to 0 decimal places:
[tex]\[ 436 \][/tex]
Therefore, the result is [tex]\(436\)[/tex] with 0 decimal places (as significant figures).
### e) [tex]\( 3.64 \times 10^{-5} \times 4.5 \times 10^9 \)[/tex]
1. Calculate the multiplication:
[tex]\[ 3.64 \times 10^{-5} \times 4.5 \times 10^9 \approx 163800.0 \][/tex]
2. Determine the number of significant figures. The operands [tex]\(3.64 \times 10^{-5}\)[/tex] and [tex]\(4.5 \times 10^9\)[/tex] have the least significant number which is 2 significant figures.
3. Round the result to 2 significant figures:
[tex]\[ 1.6 \times 10^5 \][/tex]
Therefore, the result is [tex]\(1.6 \times 10^5\)[/tex] with 2 significant figures.
### f) [tex]\( \frac{42.47 - 2.690 + \log(3.110 \times 10^2)}{(12.30 - 2.804) \times 4.2001} \)[/tex]
1. Calculate each component:
[tex]\[ 42.47 - 2.690 + \log(3.110 \times 10^2) \approx 42.47 - 2.690 + 2.493 = 42.273 \][/tex]
[tex]\[ (12.30 - 2.804) \times 4.2001 \approx 9.496 \times 4.2001 \approx 39.877896 \][/tex]
2. Perform the final division:
[tex]\[ \frac{42.273}{39.877896} \approx 1.0598887230386589 \][/tex]
3. Determine the number of significant figures. The operand [tex]\(2.804\)[/tex] has the least significant figures (3 significant figures).
4. Round the result to 3 significant figures:
[tex]\[ 1.06 \][/tex]
Therefore, the result is [tex]\(1.06\)[/tex] with 3 significant figures.
In summary, the results with the correct number of significant figures are as follows:
a) [tex]\(0.001726\)[/tex]
b) [tex]\(46\)[/tex]
c) [tex]\(0.11\)[/tex]
d) [tex]\(436\)[/tex]
e) [tex]\(1.6 \times 10^5\)[/tex]
f) [tex]\(1.06\)[/tex]
