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Question Video: Converting Scientific Notation into Decimal Form Physics • 9th Grade

A building contains 3 × 10⁵ bricks. What is the number of bricks in the building, expressed in decimal form?

05:05

Video Transcript

A building contains three times 10 to the power of five bricks. What is the number of bricks in the building expressed in decimal form?

So, in this question, we have a building. Let’s say this is our building here. And we’re told exactly how many bricks have been used to construct the building, which is three times 10 to the power of five bricks. This number is currently written in scientific notation or standard form, and this question is asking us to convert this number into its decimal form. In order to recall how to do this conversion, let’s start simply and first look at the number 10 to the power of zero.

We can recall that any number raised to the zeroth power is one. So, 10 to the power of zero equals one. Let’s now look at a slightly bigger number, 10 to the power of one. For this number, we can recall that when a number is raised to the power of one, it stays exactly the same. So, when we have 10 to the power of one, the 10 stays exactly the same and 10 to the power of one is just equal to 10. Moving on again, let’s consider 10 to the power of two or 10 squared. To square a number, we multiply it by itself. So, 10 squared is 10 times 10, which is equal to 100.

Okay, so we’ve raised 10 to the power of a few small numbers, zero, one, and two. Let’s look at the pattern that’s emerging here. Each time we add one to the power of 10 that we’re thinking about, the resultant number on the right-hand side of these equations gets an extra zero added onto the end of it. When we raise 10 to the zeroth power, we get one which has no zeros in it. When we do 10 to the first power, we get 10, which has one zero in it. And when we do 10 squared, we get 100, which has two zeros at the end of it.

The pattern that we see is that when we’re raising 10 to a power, the power tells us how many zeros to write after the one. And in fact, when we’re raising 10 to any power, this pattern continues. So, if we wanted to write down 10 to the third power or 10 cubed, we can do this easily by writing down a one followed by three zeros. And we’ve written down that 10 to the power of three is equal to 1000, which we know is correct.

Note that this pattern is only true when the number we’re raising to a power is 10. For example, let’s think about squaring a different number, say, three. We know that three squared is equal to three times three, which is nine. And nine doesn’t have any zeros in it. So, we know this can’t possibly be the same as what we would get if we applied this rule to squaring three. But now that we know how to raise 10 to different powers, let’s return to the question.

And for this question, we want to know 10 to the power of five. And for this, we can use the same pattern we’ve just seen. We can start by writing down a one on the right-hand side and follow it by the number of zeroes given to us in the power, which in this case is five. So, we’ve written down a one and we follow it by one, two, three, four, five zeroes. And in fact, we usually write in a comma for big numbers, just to make them a little bit easier to read. And this number that we’ve written down, which is equal to 10 to the power of five, is actually 100,000.

So, this gives us 10 to the power of five written in decimal notation. But we’re not quite done yet because this question asks about three times this number, three times 10 to the power of five. So, to get our final answer, we need to multiply 100,000 by three. So, let’s clear a little bit of room to do that. So, we want to work out three times 10 to the power of five, which we’ve just seen is equal to three times 100,000. Even though 100,000 is a very big number, this multiplication is pretty straightforward. And we can find that three times 100,000 is 300,000.

And so, we found what three times 10 to the power of five is in decimal form. It’s 300000 or 300,000, and this is our final answer. We found that if we write the number of bricks in the building, which is three times 10 to the power of five, in decimal form, it’s 300,000 bricks.

For questions like this, we can actually take a shortcut when writing numbers like three times 10 to the power of five in decimal form. For single digits, in this case a three, multiplied by a power of 10, in this case 10 to the power of five, the pattern is the same as we saw earlier. Only we start with the digit that’s multiplying the 10, so a three instead of a one.

This means that we can start by writing down the digit that multiplies the power of 10, in this case a three. And then we just write down the number of zeroes given to us by the power of 10, so in this case five. So, we’ve written down our three. And then we can write one, two, three, four, five zeroes after it. And this very quickly gives us the number written in decimal form, which we once again found to be 300000 or 300,000.

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