Question

How many times greater is 1, point, 6, 3, 8, times, 10, to the power minus 31.638×10
−3
than 6, point, 3, times, 10, to the power minus 56.3×10
−5
? Express your answer using either standard notation or scientific notation.

Answer 1

To find how many times greater \(1.638 \times 10^{-31.638} \) is than \(6.3 \times 10^{-56.3}\), we can divide the first value by the second value.

Let's start with the numbers in scientific notation:

1. The first number:
\[
1.638 \times 10^{-31.638}
\]

2. The second number:
\[
6.3 \times 10^{-56.3}
\]

Now, we'll perform the division:

\[
\frac{1.638 \times 10^{-31.638}}{6.3 \times 10^{-56.3}} = \frac{1.638}{6.3} \times \frac{10^{-31.638}}{10^{-56.3}}
\]

Calculating the numerical part:

\[
\frac{1.638}{6.3} \approx 0.25968
\]

Now calculating the exponent part:

Subtract the exponents:

\[
-31.638 - (-56.3) = -31.638 + 56.3 = 24.662
\]

Now combine both parts:

\[
0.25968 \times 10^{24.662}
\]

To express in scientific notation, we can convert \(0.25968\) into \(2.5968\) (which is within the range of \(1\) to \(10\)) and adjust the exponent accordingly:

\[
0.25968 = 2.5968 \times 10^{-1}
\]

Thus, we have:

\[
(2.5968 \times 10^{-1}) \times 10^{24.662} = 2.5968 \times 10^{24.662 - 1} = 2.5968 \times 10^{23.662}
\]

Therefore, the final answer to how many times greater \(1.638 \times 10^{-31.638}\) is than \(6.3 \times 10^{-56.3}\) is:

\[
\boxed{2.5968 \times 10^{23.662}}
\]