Perform the following calculations and retain the appropriate number of significant figures in each result. (a) \(\left(38.4 \times 10^{-3}\right) \times\left(6.36 \times 10^{5}\right)=\) (b) \(\frac{\left(1.45 \times 10^{2}\right) \times\left(8.76 \times 10^{-4}\right)}{\left(9.2 \times 10^{-3}\right)^{2}}=\) (c) \(24.6+18.35-2.98=\) (d) \(\left(1.646 \times 10^{3}\right)-\left(2.18 \times 10^{2}\right)+\left[\left(1.36 \times 10^{4}\right)\right.\) [Hint: The significant figure rule for the extraction of a root is the same as for multiplication.] \(\left.\times\left(5.17 \times 10^{-2}\right)\right]=\) (e) \(\frac{-7.29 \times 10^{-4}+\sqrt{\left(7.29 \times 10^{-4}\right)^{2}+4(1.00)\left(2.7 \times 10^{-5}\right)}}{2 \times(1.00)}\)

Step by step solution

01

Calculating Part (a)

Firstly, calculate the result for part (a) by multiplying the given numbers. This operation is as follows: \(38.4 \times 10^{-3} \times 6.36 \times 10^{5} = 24.38\). The result has to be rounded to the second decimal place as it contains a multiplication of two numbers and one of the numbers (38.4) contains two decimal positions.

02

Calculating Part (b)

The next operation involves the division. The calculation is \(\frac{1.45 \times 10^{2} \times 8.76 \times 10^{-4}}{9.2 \times 10^{-3}^{2}} = 152.183\). The result should be rounded up to three decimal positions because the number with the least number of decimal places in the divisor (9.2) has one significant spot behind the decimal point.

03

Calculating Part (c)

Part (c) involves an addition and subtraction process. Hence, perform the calculation: \(24.6 + 18.35 - 2.98 = 39.97\). The number of decimal positions equals the least number of decimal places in the given numbers, i.e., one (in 24.6). Hence, the result should be presented as 40.0.

04

Calculating Part (d)

In part (d), subtract and multiply the given numbers to get the result: \(1.646 \times 10^{3} - 2.18 \times 10^{2} + 1.36 \times 10^{4} \times 5.17 \times 10^{-2} = 715.94\). The result should be rounded to two decimal places as indicated by the least decimal positions number \((2.18 \times 10^{2})\).

05

Calculating Part (e)

In this final operation, take the root and carry out the additions, subtractions and division as shown: \(\frac{-7.29 \times 10^{-4} + \sqrt{\left(7.29 \times 10^{-4}\right)^{2} + 4 \times 1.00 \times 2.7 \times 10^{-5}}}{2 \times 1.00} = 0.000183\). The result should retain the same decimal positions as the number with least decimal positions in the equation. Here, after performing the square root operation, the number of decimal positions becomes 2, as in \(1.00\). Hence, round the result \((0.000183)\) up to two decimal places.

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