Step-by-step explanation:
scientific notation simply means there is one digit (unequal to 0) before the decimal point, the other digits come after the decimal point. and this is multiplied by the proper power of 10 to put this number into its correct value spot.
remember that a negative exponent means 1/...
so, 10^-1 = 1/10¹ = 1/10 or 0.1
10^-2 = 1/10² = 1/100 or 0.01
and so on.
in this exercise we just have to turn this around and find the right power of 10 for a given decimal or so.
0.000125
let's count the positions after the decimal point until the important digits begin.
right, at the 4th position after the decimal point.
so, the scientific notation is
1.25×10^-4
as when seeing this, we know that too understand the real value of that number we have take the number sequence "125" and move it to the 4th position after the decimal point.
0.0002
again, how many positions after the decimal point until the important digits start ?
right, again 4 positions.
so, the scientific notation is
2.0×10^-4
it is good style to add a 0 to provide at least one digit after the decimal point, if there are not enough digits unequal to 0.
325×10^-8
that is close to the scientific notation but violates the rule to have the first important digit before the decimal point, and the rest after the decimal point.
so, we need to transform the number sequence "325" into 3.25
remember that a number 325 is the same as 325.0
so, actually the decimal point is 2 positions too much to the right.
for 3.25 we need to bring the decimal point the 2 positions to the left.
remember that the decimal point moves one position with every multiplication or division by 10.
so, what operation do we need to move the decimal point to the left ?
to go from 325 to 32.5 - is that multiplication or division ?
right, we divided by 10.
so, divisions by 10 makes the decimal point move left (while multiplications by 10 moves it to the right).
we need to move it 2 positions to the left, so we need to divide 2 times by 10 (which is the same as dividing one time by 10×10 or 10² or 100).
but by doing this division we actually changed the value of the number. we need to counterbalance that by adjusting the multiplying power of 10.
we started with
325×10^-8
and divided the first part by 100.
325/100 = 3.25
therefore, to keep the overall value of the number the same, we need now to multiply 10^-8 by 100.
10^-8 × 100 = 10^-8 × 10^2 = 10^-6
remember, when multiplying the same numbers with exponents, we simply keep that number and add the exponents for the new exponent.
so, we actually did
325/100 × 10^-8 × 100
and the result is the correctly formatted scientific notation
3.25×10^-6
