Express the result of each of the following calculations in exponential form and with the appropriate number of significant figures. (a) \(\left(4.65 \times 10^{4}\right) \times\left(2.95 \times 10^{-2}\right) \times\left(6.663 \times 10^{-3}\right) \times 8.2=\) (b) \(\frac{1912 \times\left(0.0077 \times 10^{4}\right) \times\left(3.12 \times 10^{-3}\right)}{\left(4.18 \times 10^{-4}\right)^{3}}=\) {c} \(\left(3.46 \times 10^{3}\right) \times 0.087 \times 15.26 \times 1.0023=\) (d) \(\frac{\left(4.505 \times 10^{-2}\right)^{2} \times 1.080 \times 1545.9}{0.03203 \times 10^{3}}=\) (e) \(\frac{\left(-3.61 \times 10^{-4}\right)+\sqrt{\left(3.61 \times 10^{-4}\right)^{2}+4(1.00)\left(1.9 \times 10^{-5}\right)}}{2 \times(1.00)}\) [Hint: The significant figure rule for the extraction of a root is the same as for multiplication.]

Step by step solution

01

Calculation (a)

This one involves multiplication of four numbers. We start off the calculation by multiplying the numbers as follows: \(4.65 * 10^{4} * 2.95 * 10^{-2} * 6.663 * 10^{-3} * 8.2\). The result of this operation is 8939.71111. We then present it in exponential form; this would be \(8.93971111 * 10^{3}\). We now need to apply the rule of significant figures. In multiplication, the rule is to use the same number of significant figures in the result as in the number with the smallest number of significant figures. Here, the number with the least significant figures is 8.2 with two significant figures, so our answer should also have two significant figures. Therefore, our final answer is \(8.9 * 10^{3}\).

02

Calculation (b)

This one involves both multiplication and division. The numerator is calculated as: \(1912 * 0.0077 * 10^{4} * 3.12 * 10^{-3}\), giving 0.46211616. The denominator involves a power of \(4.18 * 10^{-4}\) cubed, which gives 0.0000000730728. Dividing the numerator by the denominator gives us \(0.46211616 / 0.0000000730728 = 6321720.03556\). We now express this in exponential form, \(6.32172003556 * 10^{6}\). The least significant figures in our numbers is three (from 1912), so our final answer should also have three significant figures. Therefore, our final answer is \(6.32 * 10^{6}\).

03

Calculation (c)

In the third calculation, we simply multiply the four participating numbers. Thus, \(3.46 * 10^{3} * 0.087 * 15.26 * 1.0023 = 4433.38214\). We change this into exponential form to give \(4.43338214 * 10^{3}\). Considering significant figures, our answer should have two significant figures (from 0.087) so our final answer is \(4.4 * 10^{3}\).

04

Calculation (d)

This problem is somewhat more complex, as we need to take the square of \(4.505 * 10^{-2}\). This gives us 0.000203025, and multiplied by 1.080 and 1545.9 gives us 0.338582174. We divide this by \(0.03203 * 10^{3}\) which gives us \(0.338582174 / 32.03 = 0.01057\), after rounding. We change this into exponential form to give \(1.057 * 10^{-2}\). Taking into account significant figures, the least significant figures are four (from 1.080), so our final answer becomes \(1.057 * 10^{-2}\).

05

Calculation (e)

This last exercise involves taking a square root. First we compute the square root of \((3.61 * 10^{-4})^2 + 4(1.00)(1.9 * 10^{-5})\) which equates to \(6.000025 * 10^{-4}\). We divide this number by \(2 * 1.00\) to have \(3.0000125 * 10^{-4}\). Our answer needs to have three significant figures because the number with the smallest number of significant figures, \(1.9 * 10^{-5}\), has two. Therefore, our final answer is \(3.00 * 10^{-4}\)

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