Finding the Unknown Base
Neil Trivedi
Teacher
Finding the Unknown Base
So far, we have covered the index rules that can be used to simplify expressions and solve equations involving powers. In this note, we will answer questions where the base is unknown, such as .
In these questions, we use index rules and work backwards to find the value of the base.
Consider a familiar example
We want to find the value of . Since is being cubed, we undo this by taking the cube root of both sides. In index form, taking the cube root is the same as raising to the power of .
Taking the cube root of both sides,
Note: Here, we are representing the cube root using indices. The rule we are using states that .
On the left-hand side, using the index rule which states that , we multiply and .
The cube root of is .
Since is simply , we have .
Let’s consider another familiar example
We want to find the value of . Since is being squared, we undo this by taking the square root of both sides. In index form, taking the square root is the same as raising to the power of , as per the rule which states that .
Taking the square root of both sides,
On the left-hand side, using the rule which states that , we multiply and .
The square root of is . However, when solving the equation , there are two solutions, one positive and one negative.
This is because both and square to give .
and
So,
It is important to note that this is different from simply evaluating a square root, such as , which will give us . The square root symbol represents the positive (principal) root only.
Now, Let’s consider a more general case
We take the root of both sides to cancel the power n which means raising to the power of .
Taking the root on both sides,
On the left-hand side, using the rule which states that , we multiply and .
So, in general, to find the unknown base, we raise both sides of the equation to the reciprocal of the power we want to remove to isolate the base.
Remember, if we have an even numerator in the power of the original equation, we will have two solutions.
Example 1:
Find in the following:
a)
Single Step: Raise both sides to the reciprocal of the power that we want to eliminate and solve for .
The reciprocal of is so we raise both sides to the power of to cancel with the on the left-hand side.
b)
Single Step: Raise both sides to the reciprocal of the power that we want to eliminate and solve for .
The reciprocal of is so we raise both sides to the power of to cancel with the on the left-hand side.
For the right-hand side, using the index rule which states that , we flip the fraction first.
Now, we take the root of .
c)
Single Step: Raise both sides to the reciprocal of the power that we want to eliminate and solve for .
The reciprocal of is so we raise both sides to the power of to cancel with the on the left-hand side.
The power of is telling us to square root the number and then cube it (see our Negative and Fractional Indices study note for more information).
Remember, we need here since in the original equation, the numerator was , which is even. It is just like how when we solve , we get two solutions.
So, we then cube both and .
If the numerator is even, we will obtain two solutions, so we must include the .
If the numerator is odd, we will obtain one solution, so we don’t need to include the .
Example 2:
Find in the following:
a)
Single Step: Raise both sides to the reciprocal of the power that we want to eliminate and solve for .
The reciprocal of is so we raise both sides to the power of to cancel with the on the left-hand side.
For the right-hand side, using the index rule which states that , we flip the fraction first.
The power of is telling us to square root the number and then cube it. So, we first root the numerator and denominator.
Like in the previous example, we need here since in the original equation, the numerator was , which is even. So, we get solutions.
We then cube both the numerator and denominator.
b)
Single Step: Raise both sides to the reciprocal of the power that we want to eliminate and solve for .
The reciprocal of is so we raise both sides to the power of to cancel with the on the left-hand side.
The power of is telling us to cube root the number and then raise it to a power of . The cube root of is .
To note, the numerator in the power of the original equation was , which is odd, so there is only one solution and the is not needed.
so,
Adding to both sides,
Practice Questions