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 for each part are: (a) \(-2333.6\) (b) \(8.2597 \times 10^7\) (c) \(33846\) (d) \(761 \times 10^3\)

Step-by-step solution

Perform Division First

According to the order of operations, we need to perform the division before the subtraction. Calculate 6104.5 divided by 2.3. \[6104.5 \div 2.3 = 2654.1304...\] When dividing, the result should have the same number of significant figures as the value with the least number of significant figures. In this case, 6104.5 has the least number of significant figures (5). Thus, our result should be rounded to 5 significant figures: \[2654.1\]

Perform Subtraction

Now, subtract the result obtained in step 1 from 320.5. \[320.5 - 2654.1 = -2333.6\] Since both values have 4 significant digits, the result should also have 4 significant digits. Thus, the final answer is: \[-2333.6\] #b) [(285.3 x 10^5) - (1.200 x 10^3)] x 2.8954#

Perform Multiplications

First, perform the two multiplications: - 285.3 x 10^5 - 1.200 x 10^3 \[285.3 \times 10^5 = 2.853 \times 10^7\] \[1.200 \times 10^3 = 1.200 \times 10^3\]

Perform Subtraction

Next, subtract the two results: \[(2.853 \times 10^7) - (1.200 \times 10^3) = 2.852998 \times 10^7\]

Perform Final Multiplication

Finally, multiply the result obtained in step 2 by 2.8954: \[(2.852998 \times 10^7) \times 2.8954 = 8.2596916 \times 10^{7}\] Our answer should have the least number of significant figures, which is 5 in this case: \[8.2597 \times 10^7\] #c) (0.0045 x 20,000.0) + (2813 x 12)#

Perform Multiplication

First, perform the multiplication 0.0045 x 20,000.0: \[0.0045 \times 20,\!000.0 = 90.0\]

Perform Another Multiplication

Now, perform the multiplication 2813 x 12: \[2813 \times 12 = 33756\]

Perform Addition

Finally, add the two results obtained in step 1 and 2: \[90.0 + 33756 = 33846\] Keep the result with the lowest decimal places, which is 1 in this case: \[33846\] #d) 863 x [1255 - (3.45 x 108)]#

Perform Multiplication

First, perform the multiplication 3.45 x 108: \[3.45 \times 108 = 372.6\]

Perform Subtraction

Now, subtract the result obtained in step 1 from 1255: \[1255 - 372.6 = 882.4\]

Perform Final Multiplication

Finally, multiply the result obtained in step 2 by 863: \[863 \times 882.4 = 761450.2\] Our answer should have the least number of significant figures, which is 3 in this case: \[761 \times 10^3\]