Calculate e-x
Enter a signed value for x. The calculator always evaluates e-x.
Example Data Table
Use these reference values to check signs and expected result sizes.
| x | Expression | Decimal value | Scientific notation | Percent of one |
|---|---|---|---|---|
| 0 | e0 | 1.000000 | 1.000000e+0 | 100% |
| 1 | e-1 | 0.367879 | 3.678794e-1 | 36.787944% |
| 2 | e-2 | 0.135335 | 1.353353e-1 | 13.533528% |
| 5 | e-5 | 0.006738 | 6.737947e-3 | 0.673795% |
| 10 | e-10 | 0.000045 | 4.539993e-5 | 0.004540% |
Formula Used
e-x = 1 / ex
The calculator applies the negative sign to x first. It then evaluates Euler’s number raised to that exponent. The percentage view uses 100 × e-x.
- e is Euler’s number, approximately 2.718281828.
- x is the signed exponent you enter.
- e-x is the reciprocal of ex.
How to Use This Calculator
- Enter the signed exponent value in the x field.
- Select how many decimal places you want to display.
- Choose decimal, scientific, or both output formats.
- Keep calculation steps selected when you need a verification trail.
- Select Calculate to place the result above the form.
- Download the CSV or print the result when documentation is needed.
Understanding e to Negative Powers
Euler’s number, written as e, is approximately 2.718281828. It appears in growth, decay, finance, probability, and engineering. A negative exponent changes growth into a reciprocal form. The expression e-x equals one divided by ex. As x increases, the result moves closer to zero. It never becomes zero for any finite x. This behavior makes exponential decay useful. It can model cooling, fading signals, radioactive change, and discount factors. The calculator removes manual rounding problems. It also gives several views of the same answer. That helps you interpret very small values.
Read the Sign Carefully
The sign in the exponent matters. For e-2, first evaluate e2. That value is about 7.389. Then take its reciprocal. The result is about 0.135335. This reciprocal relationship provides a simple accuracy check. A positive input produces a value between zero and one. A zero input produces exactly one. A negative input makes the inner negative sign reverse direction. For example, e-(-3) becomes e3. The result then exceeds one. Enter signed values carefully when a formula already contains a negative sign.
Where This Expression Appears
Exponential decay appears when change depends on the current amount. A medicine concentration may fall by a proportional rate. A capacitor may discharge through a circuit. A signal can weaken across distance. A probability model can use e-x for waiting times. In finance, continuous discount factors also use this form. Each situation needs correct units for x. Exponents should always be dimensionless. A rate and a time often multiply to create that dimensionless number. Check your source formula before copying values into the calculator.
Choose a Useful Display
The decimal format is helpful for ordinary sizes. Scientific notation is better for extremely small or large results. For example, e-20 is easier to read as 2.061154e-9. The percentage view compares the result with one. It is useful when the value represents a remaining fraction. The reciprocal check confirms the relation one divided by ex. Rounding changes displayed digits, not the internal calculation. Choose more decimal places when small differences matter. Avoid treating rounded values as exact measurements.
Use Results with Confidence
Start by entering the signed value for x. Choose the decimal precision you need. Select decimal, scientific, or both formats. Submit the form to see the result above the inputs. Review the substituted exponent and verification value. Export the result as CSV when you need a record. Use the print control to save a clean PDF copy. The examples below provide quick reference points. Compare your answer with nearby examples. That can reveal a misplaced sign or decimal. For technical work, keep enough digits until the final step. Then round only once for reporting.
Good practice combines calculation with context. Confirm that x represents the intended quantity. Preserve units outside the exponent. Record assumptions beside exported results. When comparing scenarios, use identical precision settings. This approach produces clearer reports, easier reviews, and fewer avoidable interpretation errors during routine analysis work today.
Frequently Asked Questions
1. What does e to a negative power mean?
It means the reciprocal of the matching positive power. For example, e-4 equals 1 divided by e4. The result is positive and usually less than one when the entered x value is positive.
2. What is the value of e-1?
e-1 is approximately 0.3678794412. It is also written as 3.678794412e-1. As a percentage of one, it is about 36.78794412%.
3. Can I enter a negative x value?
Yes. The calculator evaluates e-x. Entering x = -3 changes the exponent to -(-3), which equals 3. The result is e3, so it is greater than one.
4. Why does e-x become smaller as x grows?
Because ex grows as x increases. Its reciprocal therefore shrinks. The value approaches zero but does not reach zero for a finite exponent.
5. When should I use scientific notation?
Use scientific notation when the result has many leading zeros or becomes very large. It preserves scale clearly and makes values such as e-50 easier to compare.
6. Does this calculator round the internal answer?
The calculation uses the server’s floating-point exponential function. The selected precision changes the displayed result. Keep more digits during intermediate work, then round once when reporting.
7. What does the reciprocal check show?
It shows ex and confirms that e-x equals 1 divided by that value. This is a quick way to catch sign-entry mistakes.
8. Can e-x equal zero?
No. For every finite real x, e-x remains positive. It may become extremely small, but it never becomes exactly zero in mathematical terms.
9. Why is the input range limited?
The range protects the calculation from floating-point overflow and underflow. Values from -700 to 700 provide a wide practical range while keeping the exponential function dependable.
10. What units should x have?
The exponent must be dimensionless. In applied formulas, x often comes from multiplying a rate by time. Make sure the rate and time units cancel before calculating.
11. How can I avoid sign errors?
Use clear inputs, then verify results before making decisions.