Question

By what number should we multiply \[{{3}^{4}}\] so the product is \[{{3}^{7}}\]?

Answer 1

Hint: when we multiply two same numbers having different powers then the result will be the same number but their powers will be added and if we divide two same numbers having different powers then the results will be the same number but their powers will be subtracted.

Complete step by step answer:
A number can be a positive number or a negative number and its powers can also be negative or positive. If the number is having a negative power then after doing the reciprocal of that number then its power will become negative. The power of a number represents the number when it is multiplied by itself. A number consists of a base number and an exponent. The base number gives us the actual number and the number which is written in the power on the right-hand side of the base number is known as the exponent and it tells us that how many times the number will be multiplied by itself.
To calculate the power of ten is very easy. Suppose if there is ‘n’ power on ten then one will always be there and n zeros will be placed after one.
In this question, we have to find the number which should be multiplied by \[{{3}^{4}}\] to get the result \[{{3}^{7}}\].
Let that number be x, then after multiplying that number with \[{{3}^{4}}\] we get the result \[{{3}^{7}}\]which is as shown below.
\[{{3}^{4}}\times x={{3}^{7}}\]
So \[{{3}^{4}}\] will go on the right hand-side and \[{{3}^{7}}\] will be divided by \[{{3}^{4}}\] and the following results will be obtained.
\[x=\dfrac{{{3}^{7}}}{{{3}^{4}}}\]
If two numbers are the same but they are having different powers and they will be divided with each other then their powers will be subtracted from each other which is as shown below.
\[x={{3}^{7-4}}\]
\[\Rightarrow x={{3}^{3}}\]
So the value of x comes out to be \[{{3}^{3}}\].
Hence, \[{{3}^{4}}\] should be multiplied with \[{{3}^{3}}\] to get the result \[{{3}^{7}}\].

Note:
If there is any number whose power is zero then the result will always be one. If the negative number is having an odd power then the resultant number will also be negative but if the negative number will have even power then the resultant number will be a positive number.