NEGATIVE EXPONENT

Definition Of Negative Exponent

A Negative Exponent of a number equals to the reciprocal of positive exponent of the number. For instance

More About Negative Exponent

Negative exponents are used in scientific notation to designate a number smaller than one.

Video Examples: Negative exponents

Examples of Negative Exponent

Solved Example on Negative Exponent

Ques: Evaluate the expression (3 × 4)^-1.

Choices:

Solution:

Quick Summary

A negative exponent means the base is in the wrong place (numerator vs denominator).
To simplify, move the base with the negative exponent to the other part of the fraction (numerator or denominator) and make the exponent positive.
Any number (except 0) raised to the power of -1 is its reciprocal.

\[ x^{-n} = \frac{1}{x^n} \]

⚠️ Common Mistakes

Thinking a negative exponent results in a negative number (e.g., thinking 2^-2 is -4).
Forgetting to take the reciprocal of the entire base when there is a coefficient (e.g., (2x)^-1 is not 2/x).

🌍 Real-World Uses

Expressing very small numbers in scientific notation, such as the mass of an electron (approximately 9.109 × 10^-31 kg).
Calculating compound interest rates where the principal decreases over time, represented by a negative exponent in the formula for present value.
Representing the decay of radioactive isotopes using an exponential decay formula containing negative exponents, showing the decreasing amount of the isotope over time.

📋 Standards Alignment

CCSS.MATH.8.EE.A.1
CCSS.MATH.HSN-RN.A.1

🍎 Teacher Insights

Use visual aids like fraction models to show the relationship between positive and negative exponents. Emphasize that negative exponents do *not* result in negative numbers.

🎓 Prerequisites

Basic arithmetic
Understanding of fractions
Positive exponents

Check Your Knowledge

Q1: Simplify: 4^-2

16 -16 1/16 -1/16

Q2: Simplify: (1/2)^-1

1/2 -1/2 2 -2

Frequently Asked Questions

Q: What happens when the base is zero?
A: Zero raised to any negative power is undefined because it would involve division by zero.

Q: Can the exponent be a fraction?
A: Yes, negative fractional exponents are possible. For example, x^(-1/2) = 1 / sqrt(x).