Question Video: Expressing Numbers in Decimal Form Physics • 9th Grade

A block of ice is heated with 3.25 × 10⁶ J of energy. What is the number of joules of energy supplied, expressed in decimal form?

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Video Transcript

A block of ice is heated with 3.25 times 10 to the power of six joules of energy. What is the number of joules of energy supplied expressed in decimal form?

Okay, so in this question, all we’re being us to do is to take this value — 3.25 times 10 to the power of six joules, which is the amount of energy supply to our block of ice — and express it in decimal form. So what do we mean by that?

Well, any number expressed in decimal form is written in this format here. We might think of it as being written out in full. Or, in this case, a, b, c, d, e, f, g, and h can take on integer or whole number values. Where the value of f represents the number of units or ones in this number. e represents the number of tens, d represents the number of hundreds, c represents the number of thousands, and so on and so forth. b is ten thousands. a is one hundred thousands. And we can have values going on to the left here as well. And we can place values after the decimal point as well, which represent fractional values. So g represents the number of tenths there are. h represents the number of hundredths there are. And we can go on in this direction as well.

To make this clearer, let’s write down an example. How about the number 123456.78? This number is written in decimal form. And from this we can gather that in this number, 123456.78, there are six units or six ones plus five 10s, which is 50, plus four 100s, which is 400. Plus three lots of 1000, which is 3000, plus two lots of 10000, which is 20000, plus one lot of 100000 which is 100000. And, as well as this, there are seven lots of one-tenth, which is seven-tenths or 0.7, and eight lots of one hundredths, which is eight hundredths or 0.08. And so to clarify, we’re adding 100000 to 20000 to 3000 to 400 to 50 to six to seven-tenths to eight one hundredths to give us the overall value 123456.78. So that’s decimal notation.

Now, the value that we’ve been given for the energy supply to our block of ice is written in scientific notation. Where any quantity written in scientific notation is written in the form 𝑎 times 10 to the power of 𝑏. And 𝑎 is a number that can be greater than or equal to one but less than 10. And 𝑏 must be any integer or whole number. And we can clearly see that the value for the energy that we’ve been given, 3.25 times 10 to the power of six joules, is in scientific notation. We’ll ignore the unit for the moment because all we’re interested in right now is the number itself. We can see that 3.25 here is 𝑎. It’s a value between one and 10. And we can see that this number six is 𝑏. It’s an integer value, a whole number. And it’s the power to which 10 is raised.

So what we’ve been asked to do is to take this value and express it in decimal form. And to do this, all we need to realize is that 10 to the power of six simply means 10 times 10 times 10 times 10 times 10 times 10. In other words, it’s six factors of 10 multiplied together. And so, to convert our value into decimal notation, all we need to do is to multiply 3.25 by 10 six times. So if we multiply 3.25 by 10 once, we get 32.5. If we multiply it by 10 again, we get 325. If we multiply it by 10 once more, we get 3250. Doing the same thing yet again gives us 32500. Multiplying by 10 once more gives us 325000. And then, multiplying by 10 for the final time gives us 3250000.

And so, we can say that our block of ice, which is heated with 3.25 times 10 to the power of six joules of energy, can also be said to be heated with 3250000 joules of energy. This quantity and this quantity are exactly the same. They’re just expressed in different ways. When we write it like this, we’re expressing it in scientific notation. And when we write it like this, we’re expressing it in decimal form. And so, our final answer is that the number of joules of energy supply to our block of ice, expressed in decimal form, is 3250000 joules.

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