Question

(\frac{3.14 \times 10^{6}}{(7.04 \times 10^{5})(2.68 \times 10^{-4})} =)

Explanation:

Step1: Simplify the numerator and denominator separately

First, handle the coefficients and the powers of 10 in the numerator and denominator. The expression is \(\frac{3.14\times10^{6}}{(7.04\times10^{5})(2.68\times10^{-4})}\).

First, multiply the coefficients in the denominator: \(7.04\times2.68\)
\(7.04\times2.68 = 7.04\times(2 + 0.6+ 0.08)=7.04\times2+7.04\times0.6 + 7.04\times0.08=14.08+4.224+0.5632 = 18.8672\)

Then, multiply the powers of 10 in the denominator: \(10^{5}\times10^{-4}=10^{5 + (-4)}=10^{1}\)

So the denominator becomes \(18.8672\times10^{1}\)

Now, the expression is \(\frac{3.14\times10^{6}}{18.8672\times10^{1}}\)

Step2: Divide the coefficients and the powers of 10

Divide the coefficients: \(\frac{3.14}{18.8672}\approx0.1664\)

Divide the powers of 10: \(\frac{10^{6}}{10^{1}} = 10^{6 - 1}=10^{5}\)

Then multiply the results: \(0.1664\times10^{5}\)

Step3: Convert to proper scientific notation

\(0.1664\times10^{5}=1.664\times10^{4}\) (because we move the decimal point one place to the right, so we decrease the exponent of 10 by 1)

Wait, let's check the calculation again. Maybe we made a mistake in the coefficient multiplication.

Wait, let's redo the denominator multiplication:

\(7.04\times2.68\):

\(7\times2.68 = 18.76\)

\(0.04\times2.68=0.1072\)

So total is \(18.76 + 0.1072=18.8672\), that's correct.

Now, \(\frac{3.14\times10^{6}}{7.04\times10^{5}\times2.68\times10^{-4}}=\frac{3.14\times10^{6}}{(7.04\times2.68)\times(10^{5}\times10^{-4})}\)

\(10^{5}\times10^{-4}=10^{1}\), \(7.04\times2.68 = 18.8672\)

So \(\frac{3.14\times10^{6}}{18.8672\times10^{1}}=\frac{3.14}{18.8672}\times10^{6 - 1}\)

\(\frac{3.14}{18.8672}\approx0.1664\), then \(0.1664\times10^{5}=1.664\times10^{4}\) (since \(0.1664\times10^{5}=1.664\times10^{4}\))

Wait, but let's use a calculator for \(\frac{3.14}{18.8672}\):

\(3.14\div18.8672\approx0.1664\)

Then \(0.1664\times10^{5}=16640\), and \(1.664\times10^{4}=16640\), that's correct.

Alternatively, let's do the calculation step by step with more precision.

First, handle the exponents:

\(10^{6}\div(10^{5}\times10^{-4})=10^{6}\div10^{5 - 4}=10^{6}\div10^{1}=10^{5}\)

Then the coefficients: \(3.14\div(7.04\times2.68)=3.14\div18.8672\approx0.1664\)

Then \(0.1664\times10^{5}=1.664\times10^{4}\approx1.66\times10^{4}\) (or more precise)

Wait, maybe we can do the calculation as:

\(\frac{3.14\times10^{6}}{7.04\times10^{5}\times2.68\times10^{-4}}=\frac{3.14}{7.04\times2.68}\times\frac{10^{6}}{10^{5}\times10^{-4}}\)

Calculate \(\frac{10^{6}}{10^{5}\times10^{-4}}=\frac{10^{6}}{10^{1}} = 10^{5}\)

Calculate \(\frac{3.14}{7.04\times2.68}=\frac{3.14}{18.8672}\approx0.1664\)

Then \(0.1664\times10^{5}=1.664\times10^{4}\approx1.66\times10^{4}\) (or 1.66×10⁴)

Wait, but let's check with another approach. Let's multiply the denominator first:

\(7.04\times10^{5}\times2.68\times10^{-4}=(7.04\times2.68)\times(10^{5}\times10^{-4})=18.8672\times10^{1}=188.672\)

Then the numerator is \(3.14\times10^{6}=3140000\)

Then divide \(3140000\div188.672\approx16640\), which is \(1.664\times10^{4}\)

Yes, that's correct.

Answer:

\(\boxed{1.66\times10^{4}}\) (or more precisely \(1.664\times10^{4}\))