Question Video: Converting a Large Number into Decimal Notation Physics • 9th Grade

Power grid electricity is the electricity supplied to homes by power stations. It operates at a voltage of 0.0023 × 10⁵ V. What is this voltage expressed in decimal notation?

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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.

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