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Question Video: Converting a Large Number into Decimal Notation Physics • First Year of Secondary School

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A road is 0.75 × 10⁴ m long. Which of the following is equal to the length of the road? [A] 75 m [B] 750 m [C] 7500 m [D] 75000 m [E] 750000 m

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Video Transcript

A road is 0.75 times 10 to the power of four meters long. Which of the following is equal to the length of the road? (A) 75 meters, (B) 750 meters, (C) 7500 meters, (D) 75000 meters, (E) 750000 meters.

Okay, so this question is asking us about the length of a road, and let’s suppose that this orange line here represents that road. Let’s label the length of the road as 𝑙, and we’re told in the question that this length is equal to 0.75 times 10 to the power of four meters. Now, we can see that we’ve been given five different potential answer options to choose from. Each of these answers is a length in units of meters, just like the length value we were given in the question. However, the length in meters that we’ve been given, which we labeled as 𝑙, is written as a number, that’s 0.75, multiplied by a power of 10. That’s 10 to the power of four.

Unlike this, all of these answer options are written as a single integer number with no powers of 10 involved. We are asked to work out which of these five answer options is equal to the length of the road. And since we’re told that this road’s length is equal to 0.75 times 10 to the power of four meters, then what the question is asking us to do is to work out which of these five lengths is just a different way of writing the same length 𝑙 of 0.75 times 10 to the power of four meters.

To work this out, let’s begin by recalling that 10 raised to the power of four is just equal to 10 times 10 times 10 times 10. So that’s four lots of 10 multiplied together. This product of four factors of 10 then works out as 10000. Since we know that our length 𝑙 is equal to 0.75 multiplied by 10 to the power of four meters and we know that 10 to the power of four is equal to 10000, then in this expression for 𝑙, we can replace the 10 to the four by 10000. When we do this, we have that 𝑙, the length of the road, is equal to 0.75 multiplied by 10000 meters. 0.75 multiplied by 10000 works out as 7500. And so we have that 𝑙 is equal to 7500 meters.

We found then that the length of the road, which we were given as 0.75 times 10 to the power of four meters, is equal to 7500 meters. This matches the length given here in answer option (C). The length of the road is equal to 7500 meters.

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