### Video Transcript

Which of the following numbers is the greatest? Is it 5.4 multiplied by 10 to the power of six, 6.4 multiplied by 10 to the power of negative six, negative 5.4 multiplied by 10 to the power of eight, or 0.054 multiplied by 10 to the power of nine?

Before starting this question, it is important to remember a couple of key points. When multiplied by 10 to any power, we need to move all the digits that number of places. When the power is positive, as in the first, third, and fourth option, we need to move all the digits to the left, six places, eight places, and nine places, respectively. If we have a negative power, as in the second option, we move all the digits to the right. This means that the number gets smaller.

Let’s now convert each of our four options to ordinary numbers. The first option was 5.4, multiplied by 10 to the power of six. Moving the digits, five and four, six places to the left gives us an answer of 5400000. There is a quick trick we can use here to work out how many zeros will be in our answer. As the power or index was six, and there was only one number after the decimal point, we will end up with five zeros in our answer as six minus one is equal to five. The four needs to move one place to get to the units column and then will move another five places to the left, resulting in five zeros.

We can use the same method, for the third option, negative 5.4 multiplied by 10 to the power of eight. This time, the digits need to move eight places to the left. This gives us the number negative 540 million. As the index or power this time was eight, and once again there was only one number after the decimal point, we will end up with seven zeroes. Eight minus one is equal to seven. The fourth option was 0.54 multiplied by 10 to the power of nine. This time we need to move all the digits nine places to the left. There were three numbers after the decimal point in the initial number. And the index or power was nine. Nine minus three is equal to six. Therefore, 0.054 multiplied by 10 to the power of nine is 54 million.

Let’s now consider the second option where the power was negative, 6.4 multiplied by 10 to the power of negative six. As already mentioned, when the power is negative, all the digits must move to the right. Therefore, our answer is 0.0000064. The digits six and four move six places to the right. We were asked to work out which of the numbers was the greatest. As the third option gave us a negative answer, this cannot be the greatest number, as any positive number is greater than a negative number.

Clearly, the second option is also not the greatest as 0.0000064 is less than 5400000 and 54 million. We’re now left with two possible options. 5.4 multiplied by 10 to the power of six and 0.054 multiplied by 10 to the power of nine. 54 million is greater than 5400000. This means that, of the four options, the number that is the greatest 0.054 multiplied by 10 to the power of nine.