Video Transcript
Power grid electricity is the
electricity supplied to homes by power stations. It operates at a voltage of 0.0023
times 10 to the power of five volts. What is this voltage expressed in
decimal notation? (A) 23 volts, (B) 230 volts, (C)
2300 volts, (D) 2300000 volts.
Okay, so in this question, we’re
given a voltage value, which we’re told is the voltage at which power grid
electricity operates. This value that we’re given is
0.0023 times 10 to the power of five volts. The way that this number is
expressed looks kind of like scientific notation. Like with scientific notation,
we’ve got a value, in this case 0.0023, multiplied by 10 raised to an exponent,
which in this case is five.
It’s worth mentioning though that,
strictly speaking, this number is not expressed in scientific notation. A number expressed in scientific
notation is written in the form 𝑎 multiplied by 10 to the power of 𝑏, where the
value 𝑎 must be greater than or equal to one and less than 10 and the exponent 𝑏
must be an integer. So the reason that this voltage
value that we’re given is not expressed in scientific notation is that this value
here, 0.0023, is smaller than one.
To answer this question though,
whether or not this number is strictly expressed in scientific notation is not
important. What we need to do is to express
the value in decimal notation so that we can see which of these four answer options
it’s equal to. Now, decimal notation means a
single number with no powers of 10 involved. So, in this voltage value, we need
to find a way of writing it without this multiplicative factor of 10 to the power of
five.
To do this, we can begin by
recalling that 10 raised to the power of 𝑥 is the same as 𝑥 factors of 10
multiplied together. So, for example, 10 to the power of
two, or 10 squared, is equal to 10 times 10, which works out as 100. Similarly, 10 to the power of
three, or 10 cubed, is equal to 10 times 10 times 10. That’s three factors of 10, and
this works out as 1000.
When we multiply a number by 10,
that’s equivalent to moving the decimal point one space to the right. For example, if we take this value
of 0.0023 and we multiply it by 10, we get a result of 0.023. Notice that that’s exactly the same
result that we get if we take the decimal point and move it over one space to the
right. By moving the decimal point one
place to the right, 0.0023 has become 0.023, with an additional leading zero that we
can now just ignore.
Now, if multiplying a number by 10
means moving the decimal point one place to the right and 10 to the power of 𝑥 is
just 𝑥 factors of 10 multiplied together, then multiplying by 10 to the power of 𝑥
is just multiplying by 10 𝑥 times. So that’s moving the decimal point
𝑥 spaces to the right.
In this voltage that we’re given,
we’ve got the value 0.0023 multiplied by 10 to the power of five. But we know now that multiplying by
10 to the power of five is just the same as moving this decimal point five spaces to
the right. Of course, if we move the decimal
point so far to the right that it goes past the end of the digits that we’ve got,
then we need to be careful to add in any zeros that we’ll get to the left of the new
position of the decimal point. In this case, we can see that we’ve
just moved the decimal point one place past this last digit of three. So we’ve just got one zero to add
in.
So this number that we’ve got here
is the voltage expressed in decimal notation. There’s a couple more things we can
do though to write it in a more conventional way. The number has three leading
zeros. That’s a zero in the thousands
column, a zero in the ten thousands column, and a zero in the hundred thousands
column. The first nonzero digit that we’ve
got is this two in the hundreds column. We can therefore get rid of any
leading zeros to the left of this because they don’t give us any information about
the number.
The second thing we can notice is
that since we don’t have any digits to the right of this decimal point, then we
don’t actually need to write out the decimal point. We can see then that this voltage
we were given of 0.0023 times 10 to the power of five volts is equal to 230
volts. Notice that this matches the value
given here in answer option (B). The voltage of 0.0023 times 10 to
the power of five volts expressed in decimal notation is equal to 230 volts.
