Question

Carry out the following operations, and express the answer with the appropriate number of significant figures. (a) \(320.5-(6104.5 / 2.3)\) (b) \(\left[\left(285.3 \times 10^{5}\right)-\left(1.200 \times 10^{3}\right)\right] \times 2.8954\) (c) \((0.0045 \times 20,000.0)+(2813 \times 12)\) (d) \(863 \times[1255-(3.45 \times 108)]\)

Short answer

The short answers to the given problems are: (a) -2333.5 (b) \(8.261 \times 10^7\) (c) 33846 (d) 761000

Step-by-step solution

a) Calculate the division

Calculate \(6104.5 \div 2.3\). Here, the operation is division, and we need to consider the least number of significant figures from both numbers. In this case, we have 4 significant figures and 2 significant figures, respectively. Hence, the result should have 2 significant figures: \( \frac{6104.5}{2.3} = 2654 \) #a) Subtract the result from 320.5#

a) Subtract the division result from 320.5

Now subtract the result of the division from 320.5. Keep in mind that subtraction requires matching decimal places, and 320.5 has 4 significant figures (and one decimal place): \(320.5 - 2654 = -2333.5\) #b) Perform the subtraction inside the brackets#

b) Perform the subtraction inside the brackets

Calculate the subtraction inside the brackets: \( \left( 285.3 \times 10^5 \right) - \left( 1.200 \times 10^3 \right) \). For subtraction, we need to match decimal places. In order to do this, we need to rewrite both numbers with the same exponent: \[ \left( 285.3 \times 10^5 \right) - \left( 0.012 \times 10^5 \right) = (285.3 - 0.012) \times 10^5 \] Now perform the subtraction and keep 3 decimal places (lowest number of decimal places in the numbers): \( (285.288) \times 10^5 \) #b) Perform the multiplication with 2.8954#

b) Perform the multiplication with 2.8954

Now multiply that result with 2.8954, considering the lowest number of significant figures (which is 4, from 2.8954): \[ (285.288 \times 10^5) \times 2.8954 = 82610000\ \] #c) Perform the multiplication with 0.0045 and 2813#

c) Perform the multiplication with 0.0045 and 2813

Calculate the multiplications: \(0.0045 \times 20,000.0\) and \(2813 \times 12\). In both cases, consider the lowest number of significant figures from the two numbers: \(0.0045 \times 20,000.0 = 90.0\) \(2813 \times 12 = 33756\) #c) Calculate the sum#

c) Calculate the sum

Add the results of the multiplications while considering the least number of decimal places: \(90.0 + 33756 = 33846\) #d) Calculate the multiplication inside the bracket#

d) Calculate the multiplication inside the bracket

Calculate the multiplication inside the bracket: \(3.45 \times 108\). Here, consider the lowest number of significant figures which is 3 from 3.45 (note, the decimal places rule only applies for addition and subtraction operations): \(3.45 \times 108 = 372.6\) #d) Perform the subtraction inside the bracket and the final multiplication#

d) Perform the subtraction inside the bracket and the final multiplication

Now perform the subtraction inside the bracket while considering the least number of decimal places (which is 0): \[1255 - 372.6 = 882\] Finally, perform the multiplication with the least number of significant figures (which is 3 from 863): \( 863 \times 882 = 761000\) The final answers are: (a) -2333.5 (b) 82610000 (c) 33846 (d) 761000