Indices and Changing of Bases
Neil Trivedi
Teacher
Indices and Changing of Bases
We have explored several index rules that can be used to simplify expressions involving powers. Here are the rules:
Index Rules:
Rule
Rule
Rule
Rule (where cannot equal )
Rule
Rule
Rule
Rule
To apply these rules, it is important that the terms have the same base. In this note, we will practise questions where the bases are not the same at first. We will use index rules to rewrite the terms so that they all have the same base. Once the bases match, we can simplify expressions or, if we need to find unknown values in the powers, equate the powers to solve for these unknown values.
As a reminder:
This is known as the power or exponent.
This tells us how many of the base are being
multiplied together.
This is known as our base.
It is the number that’s being
multiplied repeatedly.
Consider the following example.
Find the value of such that
We cannot solve for directly by equating the powers because the bases are different. We need to rewrite so that both terms have the same base. We know that so we rewrite as .
Using the index rule which states that ,
Since the bases are the same, we can equate the powers and we get
Example 1:
Solve for in the following:
a)
Step 1: Rewrite the equation so that all terms have the same base.
We’ll rewrite the equation so that every term has a base of .
Using the rule which states that ,
We can rewrite as .
Step 2: Simplify the equation using index rules and solve for .
We know that has an invisible power of , which we should include now and then multiply the and using the rule, which states that .
Adding the powers using the rule which states that ,
Equating the powers,
b)
Step 1: Rewrite the equation so that all terms have the same base.
is not an integer power of so we need to find a number that both and are powers of. In this case, that is . Therefore, we’ll rewrite the equation so that every term has a base of .
Step 2: Simplify the equation by multiplying the powers and then solving for .
Equating the powers and solving for ,
Dividing both sides by ,
c)
Step 1: Rewrite the equation so that all terms have the same base.
First, we must notice that is a power of so we can rewrite it as .
We do not want to have in the denominator so we will use the index rule which states that to bring up the denominator. At the same time, we can rewrite the root as a power of using the rule, which states that . Therefore,
Step 2: Simplify the equation by multiplying the powers on the right-hand side and then solve for .
Equating the powers and solving for ,
Moving the to the right and the to the left,
d)
Step 1: Rewrite the equation so that all terms have the same base.
We’ll rewrite the equation so that every term has a base of and rewrite the cube root as a power of using the rule, which states that .
Step 2: Simplify the equation by multiplying all the powers in each term and then solve for .
Adding the powers using the index rule which states that ,
Equating the powers and solving for ,
Subtracting from both sides to move the smaller over to the larger one,
Dividing both sides by ,
Example 2:
Solve the following simultaneous equations:
Step 1: Rewrite each equation to form linear equations in terms of and .
First, we need to rewrite each equation so that all terms have the same base. Then, we use our index laws to simplify, if needed, and then form the linear equations by equating the powers.
For the first equation:
Simplifying the right-hand side by multiplying the powers, as per the index rule which states that ,
Expanding the brackets in the power on the right-hand side,
Equating the powers, we form a linear equation.
(equation 1)
For the second equation:
Simplifying the left-hand side by multiplying the powers, as per the index rule which states that ,
Expanding the brackets in the power on the left-hand side,
Equating the powers, we form a linear equation.
(equation 2)
Therefore, we get the following simultaneous equations.
Step 2: Solve the linear simultaneous equations.
(equation 1)
(equation 2)
Here, we will solve using substitution. We can substitute equation 1 into equation 2.
(equation 2)
Expanding the brackets on the left-hand side,
We move the to the left-hand side to collect the terms and move the to the right-hand side.
Dividing both sides by ,
Substituting the value of we found into equation 1 to obtain the value of .
(equation 1)
Therefore, the solutions are and .
Challenging Questions