Question

use a calculator to evaluate the expression. write the answer in scientific notation, c × 10ⁿ, with c rounded to two decimal places. \\(\frac{(4,610,000,000)(0.000144)}{(69,000,000)(0.000002)}\\)

Explanation:

Step1: Multiply the numerators

First, calculate the product of the numerators: \(4,610,000,000\times0.000144\times0.000002\).
First, \(4,610,000,000\times0.000144 = 4,610,000,000\times\frac{144}{1,000,000}=4,610\times144 = 663,840\).
Then, \(663,840\times0.000002=663,840\times\frac{2}{1,000,000}=\frac{1,327,680}{1,000,000} = 1.32768\).

Step2: Divide by the denominator

The denominator is \(69,000,000\). So we calculate \(\frac{1.32768}{69,000,000}\).
\(1.32768\div69,000,000 = 1.32768\times10^{-7}\div6.9\times10^{7}\) (wait, no, better to do directly: \(1.32768\div69000000 = 1.32768\div(6.9\times10^{7})=\frac{1.32768}{6.9}\times10^{-7}\approx0.192417\times10^{-7}=1.92417\times10^{-8}\)? Wait, no, I think I messed up step1. Wait, original expression: \(\frac{(4,610,000,000)(0.000144)(0.000002)}{69,000,000}\times10\). Oh! I forgot the \(\times10\) at the end. Let's redo step1 with the \(\times10\) included.

Wait, the expression is \(\frac{(4,610,000,000)(0.000144)(0.000002)}{69,000,000}\times10\).

So first, calculate numerator: \(4,610,000,000\times0.000144 = 4,610,000,000\times1.44\times10^{-4}=4,610,000,000\times1.44\times10^{-4}\). \(4,610,000,000\times10^{-4}=46100\), then \(46100\times1.44 = 46100\times1 + 46100\times0.44 = 46100+20284 = 66384\). Then multiply by \(0.000002\): \(66384\times0.000002 = 0.132768\). Now multiply by 10: \(0.132768\times10 = 1.32768\). Now divide by denominator \(69,000,000\): \(1.32768\div69,000,000\).

\(1.32768\div69000000 = 1.32768\div(6.9\times10^{7})=\frac{1.32768}{6.9}\times10^{-7}\approx0.192417\times10^{-7}=1.92417\times10^{-8}\)? Wait, no, let's use scientific notation properly.

Let's rewrite all numbers in scientific notation:

\(4,610,000,000 = 4.61\times10^{9}\)

\(0.000144 = 1.44\times10^{-4}\)

\(0.000002 = 2\times10^{-6}\)

\(69,000,000 = 6.9\times10^{7}\)

So numerator (before dividing by denominator and multiplying by 10): \((4.61\times10^{9})(1.44\times10^{-4})(2\times10^{-6})\)

Multiply the coefficients: \(4.61\times1.44\times2 = 4.61\times2.88 = 13.2768\)

Multiply the exponents: \(10^{9}\times10^{-4}\times10^{-6}=10^{9 - 4 - 6}=10^{-1}\)

So numerator (before dividing and multiplying by 10) is \(13.2768\times10^{-1}=1.32768\)

Now, the expression is \(\frac{1.32768}{6.9\times10^{7}}\times10\)

First, \(\frac{1.32768}{6.9}\approx0.192417\)

Then, \(0.192417\times10\div10^{7}=1.92417\div10^{7}=1.92417\times10^{-7}\)? Wait, no: \(\frac{1.32768\times10}{6.9\times10^{7}}=\frac{13.2768}{6.9\times10^{7}}=\frac{13.2768}{6.9}\times10^{-7}\approx1.92417\times10^{-7}\)

Now, round \(1.92417\) to two decimal places: \(1.92\) (wait, 1.92417 rounded to two decimal places is 1.92? Wait, 1.92417: the third decimal is 4, which is less than 5, so we keep the second decimal as 2. Wait, no: 1.92417, first decimal 9, second 2, third 4. So rounding to two decimal places: 1.92.

Wait, but let's check the calculation again.

Wait, original expression:

Numerator: \(4,610,000,000 \times 0.000144 = 4,610,000,000 \times 1.44 \times 10^{-4} = (4,610,000,000 \times 10^{-4}) \times 1.44 = 46100 \times 1.44 = 66384\)

Then, \(66384 \times 0.000002 = 66384 \times 2 \times 10^{-6} = 132768 \times 10^{-6} = 0.132768\)

Then, multiply by 10: \(0.132768 \times 10 = 1.32768\)

Now, divide by 69,000,000: \(1.32768 \div 69000000 = 1.32768 \div (6.9 \times 10^7) = (1.32768 \div 6.9) \times 10^{-7} \approx 0.192417 \times 10^{-7} = 1.92417 \times 10^{-8}\)? Wait, I see the mistake! 69,000,000 is \(6.9 \times 10^7\), so \(1.32768 \div 6.9 \times 10^7\) is \(1.32768 \div 69000000 = 1.32768 \div 6.9 \times 10^{-7}\). Wait, \(6.9 \times 10^7\) in the denominator, so when we divide, it's \(1.32768 \times 10^0 \div (6.9 \times 10^7) = (1.32768 \div 6.9) \times 10^{-7}\). \(1.32768 \div 6.9 \approx 0.192417\), so \(0.192417 \times 10^{-7} = 1.92417 \times 10^{-8}\). Wait, that's different from before. Where is the mistake?

Ah! Because \(4,610,000,000\) is \(4.61 \times 10^9\), \(0.000144\) is \(1.44 \times 10^{-4}\), \(0.000002\) is \(2 \times 10^{-6}\). So multiplying these: \(4.61 \times 1.44 \times 2 = 4.61 \times 2.88 = 13.2768\). Exponents: \(10^9 \times 10^{-4} \times 10^{-6} = 10^{9 - 4 - 6} = 10^{-1}\). So numerator (before dividing by denominator and multiplying by 10) is \(13.2768 \times 10^{-1} = 1.32768\). Then, the expression is \(\frac{1.32768}{6.9 \times 10^7} \times 10\). So that's \(\frac{1.32768 \times 10}{6.9 \times 10^7} = \frac{13.2768}{6.9 \times 10^7} = \frac{13.2768}{6.9} \times 10^{-7} \approx 1.92417 \times 10^{-7}\). Wait, now I'm confused. Let's do the division directly:

\(1.32768 \times 10 = 13.2768\)

\(13.2768 \div 69000000 = 13.2768 \div 69000000\)

Let's compute \(13.2768 \div 69000000\):

\(69000000 \times 1.92 \times 10^{-7} = 69000000 \times 0.000000192 = 69000000 \times 1.92 \times 10^{-7} = 69000000 \times 1.92 \times 10^{-7} = (69000000 \times 10^{-7}) \times 1.92 = 6.9 \times 1.92 = 13.248\), which is close to 13.2768. So the difference is \(13.2768 - 13.248 = 0.0288\). So \(0.0288 \div 69000000 \approx 4.17 \times 10^{-10}\). So total is \(1.92 \times 10^{-7} + 4.17 \times 10^{-10} \approx 1.92417 \times 10^{-7}\). Rounded to two decimal places, \(1.92 \times 10^{-7}\)? Wait, no, the coefficient is 1.92417, which rounded to two decimal places is 1.92. Wait, 1.92417: the third decimal is 4, so we round down, keeping the second decimal as 2. So the coefficient is 1.92, and the exponent is -7? Wait, no, let's check with a calculator:

Let's compute the original expression step by step:

First, multiply \(4,610,000,000 \times 0.000144\):

\(4,610,000,000 \times 0.000144 = 4,610,000,000 \times \frac{144}{1,000,000} = 4,610 \times 144 = 663,840\)

Then, multiply by \(0.000002\):

\(663,840 \times 0.000002 = 663,840 \times \frac{2}{1,000,000} = \frac{1,327,680}{1,000,000} = 1.32768\)

Then, multiply by 10:

\(1.32768 \times 10 = 13.2768\)

Now, divide by \(69,000,000\):

\(13.2768 \div 69,000,000 = 13.2768 \div 69000000 = 0.000000192417...\)

Now, convert to scientific notation: \(1.92417 \times 10^{-7}\)

Rounded to two decimal places, the coefficient is \(1.92\) (since the third decimal is 4, which is less than 5, we round down). So the number in scientific notation is \(1.92 \times 10^{-7}\).

Wait, but let's check with a calculator:

\(4610000000 \times 0.000144 = 4610000000 \times 0.000144 = 663840\)

\(663840 \times 0.000002 = 1.32768\)

\(1.32768 \times 10 = 13.2768\)

\(13.2768 \div 69000000 = 13.2768 / 69000000 = 0.0000001924173913\)

In scientific notation, that's \(1.924173913 \times 10^{-7}\). Rounded to two decimal places, the coefficient is \(1.92\) (because the third decimal is 4, so we don't round up the second decimal). So the answer is \(1.92 \times 10^{-7}\).

Answer:

\(1.92 \times 10^{-7}\)