1.6 K) Putting Numbers into Indices – Part 2
Another way that these questions can be made more complex is to have surds involved. The power for a square root is ½ or 0.5.
So, if we had the equation below:
We have the equation below:
This question is essentially saying “6 to the power of what gives us 36√6”. I am going to answer this question by working with the right side of the equation. Currently, the right side of the equation is saying 36 lots of √6. This means that we can add in a multiplication sign between the two terms to get:
The adding in of the multiplication sign on the right side of the equation makes it easier to work with and you will see why in a few steps time. We now need to write each of the terms (36 and √6) in the form of the base number on the left side of the equation to the power of something. The base number on the left side of the equation is 6. Therefore, we are going to write 36 and √6 as 6something; 36 when written as 6something is 62, and √6 when written as 6something is 60.5 (or 6½). This results in the right side of the equation becoming:
We have the equation:
Like the previous example, I am just going to work with the right side of the equation. The right side of the equation is saying 27 lots of √3, and this means that we can add in a multiplication sign between the two terms. This results in the right side of the equation becoming:
We now need to write each of the terms (27 and √3) in the form of the base number on the left side of the equation to the power of something. The base number on the left side of the equation is 3. Therefore, we are going to write 27 and √3 as 3something; 27 when written as 3something is 33, and √3 when written as 3something is 30.5 (or 3½). This results in the right side of the equation becoming:
We have the equation:
Before we work through this question, it is worth mentioning that there are a few different methods/ processes that we can use to find the value of f. I am going to go through two of these processes.
Process 1
On the right side of the equation we have a fraction. The first process that I am going to go through involves turning the numerator and the denominator of the fraction into the form of the base number on the left side of the equation to the power of something. The base number on the left side of the equation is 5. Therefore, we turn the numerator (125) and the denominator (√5) into the form of 5something; 125 when written as 5something is 53, and √5 when written as 5something is 50.5 (or 5½). We can now replace the numerator of the fraction with 53, and the denominator with 50.5.
The fraction is essentially saying 53 divided by 50.5, and this means that we can write the fraction as:
Process 2
The second process that I am going to go through is to split the fraction on the right side of the equation so that it looks like what is shown below:
We now write both of these terms as 5something.
The first term is 125, which when written as 5something is 53.
We now move onto placing the second term (1/√5) into the form of 5something. The second term is a fraction, and this means that we will have a negative power. The square root tells us that the power would be ½ or 0.5. We then combine this with the power being negative, which results in the power for the second term being -½ or -0.5.
The right side of the equation with both of the terms as 5something is: