LAWS OF INDICES
1. Law 1 : Multiplying
When multiplying, add the powers.
2. Law 2 : Dividing
When dividing, subtract the powers.
3. Law 3 : Powers of Powers
A power raised to another power is equivalent to multiplying the powers together.
Example 1 : Expand .
Example 2 : Simplify .
Note that the examples above are done by considering numbers and each individual letter separately.
4. Law 4 : Zero Index
By law 1, . Replacing by its value, 8, we have . This means that must equal 1. In general
5. Law 5 : Negative Index
Consider .
In general,
A special case of this is that
i.e. raising a number to the power of −1 finds its reciprocal. Also note that
i.e. raising a fraction to the power of −1 inverts the fraction.
Example1: Express as a fraction. / Example2: Express as a fraction.C1 p3 Ex 1B
6. Law 6 : Fractional Index
By law 1, . But since , we must have .
By the same token, since , we must have .
In general,
So a fractional index finds the corresponding root.
7. Rational Indices
We can use the six laws of indices to evaluate expressions involving rational indices (a rational number is one of the form ). The method is
• Deal with negative indices first. Write as .
• Then expand rational indices by writing as .
Example 1 : Evaluate .
Example 2 : Evaluate .
Example 3 : Evaluate .
Example 4 : Evaluate
.
C1 p8 Ex 1F
8. Index Equations
Some questions involve finding a missing index. In these cases, try to express both sides as powers of the same (small) number.
Example 1 : Find the value of x such that .
Comparing powers,
Example 2 : Find the value of x such that .
Example 3 : Find the value of x such that .
Example 4 : Solve the equation
We begin by writing both sides as powers of 3.
We can check this by comparing and on a calculator.
Example 5 : Solve the equation .
We begin by writing both sides as powers of 2.
Example 6 : If and calculate the values of x and y.
From the first equation,
From the second equation,
We now have a pair of simultaneous equations in x and y. We rearrange the first to get , and then substitute for x in the second equation.
Index Equations worksheet