Question

use the references to access important values if needed for this question.
it is often necessary to do calculations using scientific notation when working chemistry problems. for practice, perform each of the following calculations.
$(6.90 \times 10^{-5}) (5.74 \times 10^5) = \square$
$\dfrac{9.21 \times 10^4}{9.40 \times 10^{-4}} = \square$
$\dfrac{(7.42 \times 10^6) (8.23 \times 10^4)}{(5.74 \times 10^5) (3.84 \times 10^{-4})} = \square$
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Explanation:

First Calculation: \((6.90 \times 10^{-5})(5.74 \times 10^{5})\)

Step1: Multiply coefficients

Multiply the coefficients \(6.90\) and \(5.74\).
\(6.90\times5.74 = 39.606\)

Step2: Multiply powers of 10

Use the rule \(a^m\times a^n=a^{m + n}\) for the powers of 10. Here \(m=- 5\) and \(n = 5\), so \(10^{-5}\times10^{5}=10^{-5 + 5}=10^{0}=1\)

Step3: Combine results

Multiply the results from Step 1 and Step 2.
\(39.606\times1 = 39.606\)

Step1: Divide coefficients

Divide the coefficient \(9.21\) by \(9.40\).
\(\frac{9.21}{9.40}\approx0.9798\)

Step2: Divide powers of 10

Use the rule \(\frac{a^m}{a^n}=a^{m - n}\) for the powers of 10. Here \(m = 4\) and \(n=-4\), so \(\frac{10^{4}}{10^{-4}}=10^{4-(-4)}=10^{8}\)

Step3: Combine results

Multiply the results from Step 1 and Step 2.
\(0.9798\times10^{8}=9.798\times10^{7}\) (after adjusting to proper scientific notation)

Step1: Multiply coefficients in numerator

Multiply \(7.42\) and \(8.23\).
\(7.42\times8.23 = 61.0666\)

Step2: Multiply powers of 10 in numerator

Use \(a^m\times a^n=a^{m + n}\). For \(10^{6}\) and \(10^{4}\), we get \(10^{6 + 4}=10^{10}\)

Step3: Multiply coefficients in denominator

Multiply \(5.74\) and \(3.84\).
\(5.74\times3.84=22.0416\)

Step4: Multiply powers of 10 in denominator

Use \(a^m\times a^n=a^{m + n}\). For \(10^{5}\) and \(10^{-4}\), we get \(10^{5+(-4)} = 10^{1}\)

Step5: Divide numerator by denominator (coefficients)

Divide the result from Step 1 by Step 3.
\(\frac{61.0666}{22.0416}\approx2.7705\)

Step6: Divide numerator by denominator (powers of 10)

Use \(\frac{a^m}{a^n}=a^{m - n}\). For \(10^{10}\) and \(10^{1}\), we get \(10^{10 - 1}=10^{9}\)

Step7: Combine results

Multiply the results from Step 5 and Step 6.
\(2.7705\times10^{9}\)

Answer:

\(39.606\)

Second Calculation: \(\frac{9.21\times10^{4}}{9.40\times10^{-4}}\)