We know that a bacterium is 100 micrometers long. A virus is 1000 times smaller than the bacterium. We will use the given table to find the length of the virus in meters.
Unit of Length
Length (meter)
Decimeter
10^(- 1)
Centimeter
10^(- 2)
Millimeter
10^(- 3)
Micrometer
10^(- 6)
Nanometer
10^(- 9)
We can see that a micrometer is a 10^(- 6) meters. Let's rewrite the length of the bacterium in meters using this value.
100 micrometers = 100 * 10^(- 6) metersNext, we can evaluate the length of the bacteria in meters. First, we rewrite 100 as a power 10^2.
100 * 10^(- 6) meters = 10^2 * 10^(- 6) meters
Now, we have a product of powers with the same base. To simplify this expression we can use the Product of Powers Property.
We have the length of the bacterium in meters, and we know that the virus is 1000 times smaller than the bacterium. This means we can divide the length of bacterium by 1000 to calculate the length of the virus in meters.
10^(- 4) meters/1000 = 10^(- 4)/1000 meters
To evaluate this fraction we can rewrite 1000 as a power 10^3.
10^(- 4)/1000 meters = 10^(- 4)/10^3 meters
Finally, we have a quotient of powers with the same base. To calculate this quotient we can use the Quotient of Powers Property.
In Part A we found that a virus is 10^(- 7) meters long. Now we want to determine if this is less than, greater than, or equal to 1 micrometer. Let's again look at the given table!
Unit of Length
Length (meter)
Decimeter
10^(- 1)
Centimeter
10^(- 2)
Millimeter
10^(- 3)
Micrometer
10^(- 6)
Nanometer
10^(- 9)
We can see that 1 micrometer is 10^(- 6) meters. Now, we want to determine whether this number is greater or smaller than 10^(- 7) meters.
10^(- 6) ? 10^(- 7)
Next, we will rewrite these numbers using the Negative Exponents Property to compare them.
1/10^6 ? 1/10^7
In denominators of both fractions we have powers with the same base but with different exponents. A power with a positive base is greater when the exponent is greater.
10^6 < 10^7
Notice that when the denominator is greater, we are dividing by a larger number. As a result, the whole fraction is smaller. This means that 110^6 > 110^7 and the virus length is less than 1 micrometer.
