1.4 Negative and Fractional Indices
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Learning Objectives
- Understand and apply the laws of negative indices
- Simplify expressions involving negative indices
- Understand and apply the laws of fractional indices
- Simplify expressions involving fractional indices
- Solve equations involving negative and fractional indices
Key Concepts
Introduction to Negative Indices
A negative index indicates that the base should be taken as the reciprocal. For example, x^(-n) = 1/x^n.
This rule can be extended to any number or variable with a negative exponent.
Simplifying Expressions with Negative Indices
To simplify expressions with negative indices, rewrite the term with the negative exponent as a reciprocal.
For example, x^(-2) = 1/x^2.
Example 1
Simplify x^(-3)
Solution:
x^(-3) = 1/x^3
Example 2
Simplify 2x^(-4)
Solution:
2x^(-4) = 2/x^4
Introduction to Fractional Indices
Fractional indices are another way of expressing roots. For example, x^(1/n) represents the nth root of x, and x^(m/n) represents the nth root of x raised to the power of m.
The rule is: x^(m/n) = (n√x)^m.
Simplifying Expressions with Fractional Indices
To simplify expressions with fractional indices, first apply the root and then the power.
For example, x^(1/2) is the square root of x, and x^(3/2) is the square root of x raised to the power of 3.
Example 1
Simplify x^(1/2)
Solution:
x^(1/2) = √x
Example 2
Simplify x^(3/2)
Solution:
x^(3/2) = √x^3 = x√x
Combining Negative and Fractional Indices
You can combine negative and fractional indices by applying the rules of both simultaneously.
For example, x^(-1/2) means the reciprocal of the square root of x.
Example 1
Simplify x^(-1/2)
Solution:
x^(-1/2) = 1/√x
Example 2
Simplify x^(-3/2)
Solution:
x^(-3/2) = 1/x^(3/2) = 1/x√x
Solving Equations Involving Negative and Fractional Indices
Equations involving negative and fractional indices can be solved by first simplifying the expression, then solving the resulting equation.
Apply the inverse of the index operation to isolate the variable.
Example 1
Solve x^(-2) = 1/4
Solution:
Take the reciprocal of both sides: x^2 = 4, then take the square root of both sides: x = ±2.
Example 2
Solve x^(1/3) = 8
Solution:
Cube both sides: x = 8^3 = 512.
Test Your Knowledge
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