Negative Indices: Rules, Formulas and Solved Examples
Negative Indices show that a non-zero base with a negative exponent represents the reciprocal of the corresponding positive power. This page explains the negative exponent formula, reciprocal conversion, sign rules, fractional bases and exponent shortcuts through concise examples useful for simplification questions.
What Are Negative Indices?
For a non-zero number a and positive integer n, a⁻ⁿ means 1 divided by aⁿ. For example, 2⁻³ = 1/2³ = 1/8. In a fraction, the numerator and denominator interchange when the entire fraction has a negative exponent: (a/b)⁻ⁿ = (b/a)ⁿ, where a and b are non-zero.
Negative Indices Formula & Tricks
Important Formulas
Convert a negative exponent into a positive exponent by taking the reciprocal of the base.
Interchange the numerator and denominator, then apply the positive exponent.
A negative power in the denominator moves to the numerator and becomes positive.
When bases are equal, add the indices, including the negative sign.
Quick Tricks
A negative exponent becomes positive when its base moves from the numerator to the denominator or from the denominator to the numerator.
When a fraction is raised to a negative power, first reverse the fraction and then use the positive power.
For a negative base, an even positive exponent gives a positive result and an odd positive exponent gives a negative result. The reciprocal keeps that sign.
Negative Indices Concepts
Converting a Negative Index to a Positive Index
For a⁻ⁿ, write 1/aⁿ. The exponent applies to the base before the reciprocal is evaluated. Thus, 10⁻³ = 1/1000 and (−3)⁻² = 1/9.
Negative Indices with Fractions
Both numerator and denominator exchange positions because the entire base is reciprocated. The exponent then applies normally to the reversed fraction.
Negative Indices in Numerator and Denominator
For non-zero a, a⁻ⁿ = 1/aⁿ and 1/a⁻ⁿ = aⁿ. This rule is useful when simplifying expressions containing several powers.
Multiplication and Division of Powers
The rules are aᵐ × aⁿ = aᵐ⁺ⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Negative exponents must be included with their signs before simplifying.
Signs and Special Cases
For a non-zero base, a⁰ = 1. However, 0⁻ⁿ is undefined because it would require division by zero. For a negative base, parentheses are necessary: (−2)⁻² = 1/4, whereas −2⁻² means −(2⁻²) = −1/4.
Negative Indices Video Lessons
Watch short topic-wise lessons for quick revision.
Indices: Power, Negative and Fractional Exponents
Understand power of a power, negative exponents, and fractional exponents in indices, including the rules used to simplify expressions accurately.
Practice Negative Indices Questions
Practise published questions related to this topic.
Negative Indices Quick Quiz
Attempt 5 questions and check your score instantly.
Keep practising
Practice more Negative Indices questions in the PrepShots app and continue from your current topic.
Practice More Questions - Start ₹1 Trial →Quick Revision Notes
Negative Indices: Quick Revision
Use these rules to simplify powers with negative exponents.
- a⁻ⁿ = 1/aⁿ for a ≠ 0.
- A negative exponent means reciprocal, not a negative value.
- (a/b)⁻ⁿ = (b/a)ⁿ for non-zero a and b.
- A negative power in the denominator becomes a positive power in the numerator.
- For equal bases, multiply by adding exponents and divide by subtracting exponents.
- a⁰ = 1 for a ≠ 0; 0⁻ⁿ is undefined.
- Use parentheses for negative bases when the exponent applies to the complete base.
Negative Indices FAQs
What is the negative exponent formula?
For a non-zero base, a⁻ⁿ = 1/aⁿ. For example, 4⁻² = 1/4² = 1/16.
How do you simplify (3/7)⁻²?
Reverse the fraction and apply the positive exponent: (3/7)⁻² = (7/3)² = 49/9.
What is the value of 1/5⁻³?
The negative power moves to the numerator and becomes positive: 1/5⁻³ = 5³ = 125.
Is a negative exponent equal to a negative number?
No. A negative exponent represents a reciprocal. For example, 2⁻³ = 1/8, which is positive.
What is the value of (−2)⁻⁴?
(−2)⁻⁴ = 1/(−2)⁴ = 1/16 because the even power makes the denominator positive.
What is the value of (−2)⁻³?
(−2)⁻³ = 1/(−2)³ = −1/8 because the odd power keeps the denominator negative.