What an Exponent Means, and How to Evaluate One
xⁿ means x multiplied by itself n times. 2¹⁰ = 1,024. Beyond whole numbers, the same idea extends smoothly: a negative exponent means a reciprocal (5⁻³ = 1/125), a zero exponent gives 1 and a fraction means a root.
| Expression | Meaning | Value |
|---|---|---|
| 2¹⁰ | 2 × 2 × … ten times | 1,024 |
| (−3)⁴ | four negatives | 81 |
| 5⁻³ | 1 ÷ 5³ | 1/125 |
| (2/3)⁻² | flip, then square | 9/4 |
| 8^(2/3) | (∛8)² | 4 |
| 16^(3/4) | (⁴√16)³ | 8 |
The calculator keeps results exact whenever it can. Very large results such as 2¹⁰⁰ switch to scientific notation (1.267651 × 10³⁰) and show the exact digits on request.
The Laws of Exponents
| Law | Example |
|---|---|
| xᵃ · xᵇ = xᵃ⁺ᵇ | x⁵ · x³ = x⁸ |
| xᵃ / xᵇ = xᵃ⁻ᵇ | x⁷ / x² = x⁵ |
| (xᵃ)ᵇ = xᵃᵇ | (x²)³ = x⁶ |
| (xy)ᵃ = xᵃ yᵃ | (3x²y)³ = 27x⁶y³ |
| x⁻ᵃ = 1 / xᵃ | x⁻² = 1/x² |
| x^(a/b) = ᵇ√(xᵃ) | x^(3/4) = ⁴√(x³) |
The Simplify tab applies these in order and tells you which law was used each time. For (a³b²)⁴ ÷ (a²b)³ you get a⁶b⁵.
Negative and Fractional Exponents
A negative exponent does not make the number negative. It moves the base across the fraction bar: x⁻² is 1/x², and (x⁻²y³)⁻¹ is x²/y³. For fractional exponents, the denominator is the root and the numerator is the power, so 27^(2/3) = (∛27)² = 9. Take the root first: it keeps the numbers small.
Scientific Notation
Write a number as m × 10ⁿ with 1 ≤ m < 10. Then 93,000,000 = 9.3 × 10⁷ and 0.000052 = 5.2 × 10⁻⁵. Multiplying adds exponents: (3 × 10⁸)(2.5 × 10⁻³) = 7.5 × 10⁵. Dividing subtracts: (6.4 × 10⁵) ÷ (1.6 × 10²) = 4 × 10³. For addition, match the exponents first: 4.2 × 10⁴ + 0.31 × 10⁴ = 4.51 × 10⁴. You can also round to a chosen number of significant figures, such as 1/3 to 4 s.f. = 3.333 × 10⁻¹.
Solve Exponential Equations
5 · 2ˣ = 100
- Divide by 5: 2ˣ = 20.
- x = log₂ 20 = ln 20 ÷ ln 2 ≈ 4.3219.
Answer: x ≈ 4.3219
Compound growth is the practical version: how long for $1,000 at 5% to double? Solve 1.05ˣ = 2, so x = ln 2 ÷ ln 1.05 ≈ 14.21 years. The "Solve" tab handles this form directly.
Logarithms as the Inverse of Exponents
log_b(y) asks "what power of b gives y?" So log₂ 32 = 5, log₁₀ 0.001 = −3 and log₈ 4 = 2/3 (since 8^(2/3) = 4). When the log is not a neat number, you get a decimal: log₃ 20 ≈ 2.7268 and ln 10 ≈ 2.3026.
Powers Tables for Quick Reference
The Table tab prints powers of any base. Powers of 2 up to 2¹⁰: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024. Powers of 3: 3, 9, 27, 81, 243, 729. They are worth memorising for mental maths and for computer-science work, where every byte count is a power of two.
Real Numbers: How Fast Powers Grow
| Power | Value | What it shows |
|---|---|---|
| 2¹⁰ | 1,024 | The origin of the kilobyte |
| 2³⁰ | 1,073,741,824 | Doubling 30 times passes a billion |
| 2⁶⁴ − 1 | 18,446,744,073,709,551,615 | Grains of rice on a chessboard, doubling each square |
| 2⁻³ | 0.125 | A negative exponent is a reciprocal |
| 9^(1/2) | 3 | A fractional exponent is a root |
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: memorising a list of exponent laws without understanding them. Every law follows from “a power is repeated multiplication”.
Underrated: writing a negative or fractional exponent as a reciprocal or root first. It removes most sign errors.
Plain wrong: the claim that (a + b)² = a² + b². It equals a² + 2ab + b². At a = 3, b = 4 that is 49, not 25.
Practice with Answers
- 2⁻⁴
Answer
1/16 - (2x³)⁴
Answer
16x¹² - x⁵ · x⁻²
Answer
x³ - (8x⁶)^(2/3)
Answer
4x⁴ - Solve 5 · 2ˣ = 100
Answer
x ≈ 4.32 - (4 × 10⁵)(3 × 10⁻²)
Answer
1.2 × 10⁴
Why Students Get Stuck on Exponents, and What a Good Calculator Should Do
Searching for help with exponents usually starts late at night with one problem that will not work out. The laws are simple, but negative, zero and fractional powers cause slips. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to apply each exponent law and simplify with confidence, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Adding exponents when bases are multiplied but also when addedSimplify (2x^3)^2 * x^4↗
- Treating (a + b)² as a² + b²Evaluate 8^(2/3)↗
- Misreading a negative exponent as a negative numberSimplify x^5 / x^2↗
- Not knowing what a fractional power meansEvaluate 2^(-3)↗
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants the law named for each step
- Wants fractional and negative powers explained
- Wants a decimal value to check
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.Simplify (2x^3)^2 * x^4↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.