There are three parts to this question. Part a) Calculate 1.8 multiplied by 0.003. Part b) Write 0.0000094 in standard form. Part c) Write 3.25 multiplied by 10 to the power of seven as an ordinary number.
The first part of the question asks us to multiply two decimals, 1.8 and 0.003. There are lots of ways of multiplying two decimals. One way is to ignore the decimal points and multiply two integers first. In this case, ignoring the decimal point in our first number would leave us with 18 and ignoring the decimal point in the second number would leave us with three. Let’s first multiply 18 and three.
10 multiplied by three is equal to 30 and eight multiplied by three is equal to 24. As 10 plus eight is equal to 18, we can calculate 18 multiplied by three by adding 30 and 24. 30 plus 24 equals 54. Therefore, 18 multiplied by three is equal to 54. Our first number was actually 1.8 and not 18. To get from 18 to 1.8, we divide by 10. Our second number was 0.003. To get from three to 0.003, we divide by 1000. This is because the digit three has moved from the units column to the thousandths column. It has moved three places to the right.
Our first number has been divided by 10 and our second number has been divided by 1000. 10 multiplied by 1000 is equal to 10000. This means that in order to calculate 1.8 multiplied by 0.003, we need to divide the answer of 18 multiplied by three by 10000. Dividing 54 by 10000 gives us 0.0054. The digits five and four have moved four places to the right. The five has gone from the tens to the thousandths column and the four has gone from the units to the ten thousandths column. 1.8 multiplied by 0.003 is equal to 0.0054.
The second part of our question asked us to write 0.0000094 in standard form. A number is in standard form if it is written 𝑎 multiplied by 10 to the power of 𝑛, where 𝑎 must be greater than one and less than 10. The power 𝑛 must be an integer. It is a positive integer for very large numbers and a negative integer for very small numbers.
The first nonzero digit in our number is a nine. This must be the only digit before the decimal point. As the only other digit in our number is four, the number that corresponds to 𝑎 in this case is 9.4. 9.4 is greater than one, but less than 10. We know that this number 9.4 must be multiplied by 10 to some power. If our original number is less than one and in this case a very small number, the power must be negative. This is because to get from 9.4 to the initial number 0.0000094, we would actually need to divide by 10.
Multiplying by 10 to the power of negative two is the same as dividing by 10 squared or 10 to the power of two. When writing a number in standard form, the sign must always be a multiplication one. To get from its original position to the units column, the nine has moved six places to the left. This means that the power 𝑛 is equal to negative six. The number 0.0000094 written in standard form is 9.4 multiplied by 10 to the power of negative six.
The third and final part of the question asked us to write 3.25 multiplied by 10 to the power of seven as an ordinary number. 10 to the power of seven is the same as 10 multiplied by 10 multiplied by 10 multiplied by 10 multiplied by 10 multiplied by 10 multiplied by 10. There are seven 10s. Every time we multiply a number by 10, we move every digit one place to the left. Therefore, multiplying a number by 10 to the power of seven would move all of the digits seven places to the left.
The digits three, two, and five need to move seven places to the left when multiplying by 10 to the power of seven. When we move each of the digits seven places to the left, we’re left with 32500000. The number in standard form 3.25 multiplied by 10 to the power of seven written as an ordinary number is 32500000.
There is a quick trick we can use to work out how many zeros our number will have as an ordinary number. Firstly, we can say that the power of 10, in this case seven, and the number of digits after the decimal point, in this case two and we subtract the two numbers. Seven minus two is equal to five. This means that we will have five zeros in our answer. We will have the digits three, two, and five followed by five zeros. This is a good check to use anytime we need to write a number in standard form as an ordinary number.