Question

an eon is a measure of geologic time that is equivalent to one billion years. sharks have been on earth for approximately $4.5 \times 10^{-1}$ eons. squirrels have been on earth for approximately $3.6 \times 10^{-2}$ eons. how much longer have sharks been on earth than bears? give your answer in scientific notation. $\square \times 10^{\square}$ eons

Explanation:

Step1: Identify the values

Sharks' time: $4.5 \times 10^{-1}$ eons, Squirrels' (assuming "bears" is a typo for "squirrels") time: $3.6 \times 10^{-2}$ eons.

Step2: Subtract the two values

First, rewrite $4.5 \times 10^{-1}$ as $45 \times 10^{-2}$ (to have the same exponent). Then subtract: $45 \times 10^{-2}- 3.6 \times 10^{-2}=(45 - 3.6)\times 10^{-2}=41.4\times 10^{-2}$.

Step3: Convert to proper scientific notation

$41.4\times 10^{-2}=4.14\times 10^{-1}$ (since $41.4 = 4.14\times 10^{1}$, so $4.14\times 10^{1}\times 10^{-2}=4.14\times 10^{-1}$). Wait, correction: Wait, $4.5\times 10^{-1}=0.45$, $3.6\times 10^{-2}=0.036$. Subtract: $0.45 - 0.036 = 0.414$. Convert to scientific notation: $4.14\times 10^{-1}$? Wait no, $0.414 = 4.14\times 10^{-1}$? Wait $10^{-1}$ is 0.1, $4.14\times 0.1 = 0.414$. Yes. Wait, but let's do it with exponents properly. $4.5\times 10^{-1}-3.6\times 10^{-2}=4.5\times 10^{-1}-0.36\times 10^{-1}=(4.5 - 0.36)\times 10^{-1}=4.14\times 10^{-1}$? Wait no, $3.6\times 10^{-2}=0.36\times 10^{-1}$, so $4.5\times 10^{-1}-0.36\times 10^{-1}=(4.5 - 0.36)\times 10^{-1}=4.14\times 10^{-1}$? Wait but $4.14\times 10^{-1}=0.414$, and $0.45 - 0.036 = 0.414$, which matches. So the difference is $4.14\times 10^{-1}$? Wait no, wait $4.5\times 10^{-1}$ is 0.45, $3.6\times 10^{-2}$ is 0.036. 0.45 - 0.036 = 0.414. In scientific notation, that's $4.14\times 10^{-1}$? Wait, no: $0.414 = 4.14\times 10^{-1}$? Wait $10^{-1}$ is 0.1, so 4.14 * 0.1 = 0.414. Yes. Alternatively, $41.4\times 10^{-2}=4.14\times 10^{-1}$ (because $41.4\times 10^{-2}=4.14\times 10^{1}\times 10^{-2}=4.14\times 10^{-1}$). So the calculation is correct.

Answer:

$4.14 \times 10^{-1}$ eons (Wait, but let's check again. Wait, $4.5\times 10^{-1}=0.45$, $3.6\times 10^{-2}=0.036$. 0.45 - 0.036 = 0.414. 0.414 in scientific notation is $4.14\times 10^{-1}$? Wait, no: $10^{-1}$ is 0.1, $4.14\times 0.1 = 0.414$. Yes. So the answer is $4.14\times 10^{-1}$? Wait, but maybe I made a mistake in the exponent. Wait, $0.414 = 4.14 \times 10^{-1}$, because when you move the decimal one place to the right, you multiply by $10^{-1}$. Yes. So the first box is 4.14, the second is -1. Wait, but let's re - calculate:

$4.5\times 10^{-1}=4.5\times\frac{1}{10}=0.45$

$3.6\times 10^{-2}=3.6\times\frac{1}{100}=0.036$

Subtraction: $0.45 - 0.036 = 0.414$

To write 0.414 in scientific notation:

We need to have a number between 1 and 10 multiplied by a power of 10.

$0.414 = 4.14\times 0.1=4.14\times 10^{-1}$ (since $0.1 = 10^{-1}$)

So the answer is $4.14\times 10^{-1}$ eons.