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​​1.6 I) Raising a Power to Another Power – Hard
​Whenever we raise a power to another power, we multiply the powers. We are going to be looking at a few different types of examples in this section whereby we raise a power to another power. The process used to answer each of the different types of questions is broadly the same, but there are some slight differences. Let’s have a look at some examples.
Example 1
a and b in the equation below are positive integers.

Work out the values of a and b.

 

On the left side of the equation, we have a power to another power. We can take the expression out of the bracket by multiplying all of the powers inside the bracket by the power that is on the outside of the bracket (3). Before we multiply the powers on the inside of the bracket by the power on the outside of the bracket, I am going to add in a power of 1 for the unknown a; this results in a becoming a1. The adding in of the power of 1 to a is to ensure that I do not forget to multiply any of the powers inside the bracket by the power that is on the outside of the bracket. The adding in of a power of 1 for a results in the equation becoming:

We now multiply all of the powers on the inside of the bracket by the power that is on the outside of the bracket; we multiply all of the powers inside the bracket by 3.

The next step is to equate parts of the expressions on the left and right side of the equation. We can equate the number part of the expressions and the unknown (x) part of the expressions. When we equate the number part of the expressions, we see that a3 and 8 are equal to each other. Also, when we equate the unknown part of the expressions, we see that x3b and x15 are equal to one another. This gives us the two equations that are shown below:

We can now solve these equations to find the values of a and b. Both of these equations only have one unknown in them (ignoring x as an unknown), and this means that we can solve the equations to find the values of a and b in any order.

 

Let’s solve the number equation first to find the value of a. This equation is shown below:

We want to find the value of a and not a3. Therefore, we cube root both sides of the equation.

This tells us that a is 2.

 

Let’s now solve the unknown (x) equation to find the value of b. This equation is shown below:

Both sides of the equation have a base of x, and this means that the powers must be the same; 3b must be equal to 15. Therefore, we can get rid of the bases on both sides of the equation to just leave the powers. The equation becomes:

We want to find the value of b and not 3b. Therefore, we divide both sides of the equation by 3.

This tells us that b is 5.

 

We have now found the values of both a and b; a is 2 and b is 5.

Example 2
We are now going to look at another example that is very similar to the first example. The only difference with this example is that the unknowns that we are looking for will be in different positions.
 
c and d in the equation below are positive integers.

Work out the values of c and d.

 

Like the previous example, the first step in answering this question is to take the expression on the left side of the equation out of the bracket, which we do by multiplying all of the powers on the inside of the bracket by the power that is on the outside of the bracket. Before we multiply the powers, I am going to add in a power of 1 for the 3; the 3 becomes 31.

We now multiply all of the powers on the inside of the bracket by the power that is on the outside of the bracket; we multiply all of the powers inside the bracket by c.

The next step is to equate parts of the expressions on the left and right side of the equation. We can equate the number part of the expressions and the unknown (y) part of the expressions. When we equate the number part of the expressions, we see that 3c and 81 are equal to each other. Also, when we equate the unknown part of the expressions, we see that y6c and yd are equal to one another. This gives us the two equations that are shown below:

The equation that we obtained from equating the unknowns (y) has two unknowns in it; the equation has both c and d in it. The presence of the two unknowns means that we are unable to solve this equation. Therefore, we need to solve the equation that we obtained from equating the numbers first because there is only one unknown in this equation. The equation from the numbers is shown below:

The equation here is essentially saying “3 to the power of what is 81”, where the what is c. The easiest way to find the power (c) is to keep multiplying 3 by itself until we get 81 (this technique is explored in a different section; click here to be taken to that section). The working is shown below:

From the above working, we can see that 3 has been multiplied 4 times. This means that c is 4.

 

We now need to find the value of d, which we do by subbing c as 4 into the equation that we obtained from equating the unknowns (y). The equation from equating the unknowns is shown below:

We now sub in c as 4.

Both of the bases for the two terms above are y. This means that the powers must be the same. Therefore, d must be 24.

 

We now have the values of both of the unknowns; c is 4 and d is 24.