What does 20 mean?
It means the square root of 144 plus two cubed: √144 is 12 and 2³ is 8, so 12 + 8 is 20. The expression √144 + 2³ has two operations, and the order matters — roots and powers are worked out before addition, so the 12 and the 8 are settled first and the answer is 20. A scientific calculator is a basic calculator with functions added: trig, logs, powers, roots and factorials, all in one expression, with parentheses to control the order.
Three things to check before you trust the number
Degrees or radians — pick the right one before you type. sin(30°) is 0.5 in degrees mode, but sin(30) in radians mode is −0.988 — the same keystrokes, a different angle unit, a different answer. Most school and everyday work uses degrees; calculus-style work uses radians. If a trig answer looks wrong, the mode is the first thing to check.
Functions bind tightest. √144 + 2³ is 20 because the root and the cube are worked out before the addition — and sin(30° + 15°) is sin(45°), which is 0.707107, not sin(30°) + 15°. Parentheses decide what a function applies to: sin(30°) + 15° would be a different sum entirely. When in doubt, put the function's argument in brackets.
Some inputs have no answer. √(−1) is not a real number, log(0) is not defined, and tan(90°) has no value — the calculator says so instead of printing a huge float or a made-up number. The same honesty applies to a factorial of 3.5, an empty expression, and a missing closing bracket: there is nothing to work out, so the calculator asks instead of guessing.
Worked example
On the defaults: work out √144 + 2³, then sin(30°).
- Find the root: √144 is 12, because 12 × 12 is 144.
- Find the cube: 2³ is 2 × 2 × 2, which is 8.
- Add: 12 + 8 is 20.
- Check it backwards: 20 − 8 is 12, and 12² is 144, so the root checks out.
- The trig example: in degrees mode, sin(30°) is 0.5.
- Check the cosine: cos(60°) is 0.5 too — sine and cosine swap at 30° and 60°.
- Check the tangent: tan(45°) is 1.
- Check the log: log(100) is 2, because 10² is 100.
- Check the natural log: ln(e) is 1, because e¹ is e.
- Check the factorial: 5! is 5 × 4 × 3 × 2 × 1, which is 120.
- Check the power: 2^10 is 1024.
- Check the root: √2 is 1.414214 to six places — it never terminates.
- Check pi: π is 3.141593 to six places.
Every branch, checked
- √144 + 2³: √144 + 2³: √144 + 2³ = 20. √144 + 2³ = 20 — the square root of 144 is 12 and 2 cubed is 8, so 12 + 8 is 20.
- sin(30°): sin(30°): sin(30°) = 0.5. sin(30°) = 0.5 — in degrees mode, the sine of thirty degrees is exactly one half.
- cos(60°): cos(60°): cos(60°) = 0.5. cos(60°) = 0.5 — the cosine of sixty degrees is exactly one half.
- tan(45°): tan(45°): tan(45°) = 1. tan(45°) = 1 — the tangent of forty-five degrees is exactly one.
- log(100): log(100): log(100) = 2. log(100) = 2 — the base-ten log of 100 is 2, because 10² is 100.
- ln(e): ln(e): ln(e) = 1. ln(e) = 1 — the natural log of e is 1, because e¹ is e.
- 5!: 5!: 5! = 120. 5! = 120 — five factorial is 5 × 4 × 3 × 2 × 1, which is 120.
- 2^10: 2^10: 2^10 = 1024. 2^10 = 1024 — two to the tenth power is 1024, the number behind the kilobyte.
- √2: √2: √2 = 1.414214. √2 = 1.414214 — the square root of 2 never terminates, so the display stops at six decimals.
- π: π: π = 3.141593. π = 3.141593 — pi to six decimal places.
More values the same model gives
- 1/3: 1/3 = 0.333333 — 1/3 = 0.333333 — one third never terminates, so the display stops at six decimals.
- e^2: e^2 = 7.389056 — e² = 7.389056 — e squared, to six decimal places.
- 3^3: 3^3 = 27 — 3³ = 27 — three cubed is 27.
- 1/4: 1/4 = 0.25 — 1/4 = 0.25 — one quarter is exactly 0.25.
- 100 − 25 × 3: 100 − 25 × 3 = 25 — 100 − 25 × 3 = 25 — multiplication first: 25 × 3 is 75, and 100 − 75 is 25.
The reverses, worked backwards
- Smallest whole number for n² to reach 100: 10 — at 10 the figures are 10^2 = 100 (100.000000 unrounded); at the lower neighbor the figures are 9^2 = 81; at the upper neighbor the figures are 11^2 = 121 — verdict: STEP.
- Smallest whole number for 2ⁿ to reach 1024: 10 — at 10 the figures are 2^10 = 1024 (1024.000000 unrounded); at the lower neighbor the figures are 2^9 = 512; at the upper neighbor the figures are 2^11 = 2048 — verdict: STEP.
- Largest whole number for n! to stay at 120 or below: 5 — at 5 the figures are 5! = 120 (120.000000 unrounded); at the lower neighbor the figures are 4! = 24; at the upper neighbor the figures are 6! = 720 — verdict: STEP.
- Smallest whole number for √n to reach 10: 100 — at 100 the figures are √100 = 10 (10.000000 unrounded); at the lower neighbor the figures are √99 = 9.949874; at the upper neighbor the figures are √101 = 10.049876 — verdict: STEP.
- Largest whole number for log(n) to stay at 2 or below: 100 — at 100 the figures are log(100) = 2 (2.000000 unrounded); at the lower neighbor the figures are log(99) = 1.995635; at the upper neighbor the figures are log(101) = 2.004321 — verdict: STEP.
- Smallest whole number for n³ to reach 1000: 10 — at 10 the figures are 10^3 = 1000 (1000.000000 unrounded); at the lower neighbor the figures are 9^3 = 729; at the upper neighbor the figures are 11^3 = 1331 — verdict: STEP.
What the defaults assume
The defaults are the expression √144 + 2³, which is 20, in degrees mode. They are a starting point, not a standard: your sums may be bigger or smaller, positive or negative, in degrees or radians, and the right answer for you is the one your numbers describe. Tap the keys or type your own expression and the calculator recomputes instantly. Put a sum in, read the answer, and check the working before you use it.
These notes repeat in every explainer section because this page is built to be read start to finish: pick degrees or radians before you type; functions bind tightest, so bracket their arguments; and some inputs have no answer, so the calculator says so.
Output · formula · meaning
| Output | Formula | What it means |
|---|
| Expression | input | √144 + 2³. |
| Result | √ and ^ before + | 20. |
| Result (6dp unrounded) | unrounded to 6dp | 20.000000. |
| Angle mode | degrees by default | Degrees (DEG). |
| Verdict | in words | √144 + 2³ = 20. |
| Reverse: Smallest whole number for n² to reach 100 | step | 10: 10^2 = 100. |
| Reverse: Smallest whole number for 2ⁿ to reach 1024 | step | 10: 2^10 = 1024. |
| Reverse: Largest whole number for n! to stay at 120 or below | step | 5: 5! = 120. |
| Reverse: Smallest whole number for √n to reach 10 | step | 100: √100 = 10. |
| Reverse: Largest whole number for log(n) to stay at 2 or below | step | 100: log(100) = 2. |
| Reverse: Smallest whole number for n³ to reach 1000 | step | 10: 10^3 = 1000. |
| √144 + 2³ | same model | √144 + 2³ = 20. |
| sin(30°) | same model | sin(30°) = 0.5. |
| cos(60°) | same model | cos(60°) = 0.5. |
| tan(45°) | same model | tan(45°) = 1. |
| log(100) | same model | log(100) = 2. |
| ln(e) | same model | ln(e) = 1. |
| 5! | same model | 5! = 120. |
| 2^10 | same model | 2^10 = 1024. |
| √2 | same model | √2 = 1.414214. |
| π | same model | π = 3.141593. |
| 1/3 | same model | 1/3 = 0.333333. |
| e^2 | same model | e^2 = 7.389056. |
| 3^3 | same model | 3^3 = 27. |
| 1/4 | same model | 1/4 = 0.25. |
| 100 − 25 × 3 | same model | 100 − 25 × 3 = 25. |
| Division by zero | same model | Cannot divide by zero.. |
| Square root of a negative number | same model | Square root of a negative number.. |
| log(0) | same model | Logarithm of zero or a negative number.. |
| ln(−5) | same model | Natural log of zero or a negative number.. |
| tan(90°) | same model | Tangent is undefined at this angle.. |
| Factorial of a non-integer | same model | Factorial is defined for whole numbers only.. |
| 0! | same model | 0! = 1. |
| Empty expression | same model | Enter an expression.. |
| Mismatched parentheses | same model | Missing closing parenthesis.. |
| 0.1 + 0.2 | same model | 0.1 + 0.2 = 0.3. |
| 1 ÷ 7 | same model | 1 ÷ 7 = 0.142857. |
Who uses this
- A student checking trig homework — sin(30°) is 0.5, and the degrees note explains why the same keystrokes give −0.988 in radians mode.
- An engineering student working with logs and powers — log(100) is 2 and 2^10 is 1024, with the working shown for each.
- A teacher demonstrating order of operations — √144 + 2³ is 20, and the worked example shows exactly where the 12 and the 8 come from.
- A hobbyist sizing a project — √2 is 1.414214 and 5! is 120, with six-decimal display for the numbers that never terminate.
Accuracy & limitations
Plain arithmetic on this calculator uses exact fraction arithmetic — numerator and denominator pairs, reduced to lowest terms, never floats — so 0.1 + 0.2 is exactly 0.3, not 0.30000000000000004, and 100 − 25 × 3 is exactly 25. Functions (sin, cos, tan, log, ln, √ of a non-square, eˣ) are computed as ordinary floating-point numbers and shown to six decimal places: √2 is 1.414214, π is 3.141593, e² is 7.389056, and 1 ÷ 3 is 0.333333. Trig uses degrees by default — switch to radians for calculus-style work. The calculator refuses √(−1), log(0), ln of a negative, tan(90°) and a factorial of a non-integer instead of printing a misleading number, and it refuses an expression with a missing bracket. What it cannot do is symbolic algebra, inverse trig, or complex numbers — those belong on a graphing calculator.
Cite this calculator
√144 + 2³ = 20. sin(30°) = 0.5. cos(60°) = 0.5. tan(45°) = 1. log(100) = 2. ln(e) = 1. 5! = 120. 2^10 = 1024. √2 = 1.414214. π = 3.141593. Smallest whole number for n² to reach 100: 10. Smallest whole number for 2ⁿ to reach 1024: 10. Largest whole number for n! to stay at 120 or below: 5. Smallest whole number for √n to reach 10: 100. Largest whole number for log(n) to stay at 2 or below: 100. Smallest whole number for n³ to reach 1000: 10. Source: Bright Side Kit, brightoolbox.com.
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Frequently asked questions
How do you work out √144 + 2³?
The root is 12 and the cube is 8, so 12 + 8 is 20. Type the expression into the calculator above, or tap √, 144, +, 2, x³ and =, and the answer appears as you type.
Why does sin(30°) give 0.5 but sin(30) in radians gives −0.988?
Because the angle unit changed: 30° is one twelfth of a circle, while 30 radians is almost five full circles. Check the DEG/RAD toggle before you type — degrees for school and everyday work, radians for calculus.
What is the difference between log and ln?
log is base ten and ln is base e: log(100) is 2 because 10² is 100, and ln(e) is 1 because e¹ is e. Use log for decibels, pH and engineering scales, and ln for growth, decay and calculus.
Why does tan(90°) give an error instead of a huge number?
Because the tangent of 90° is undefined — the cosine underneath it is zero, and dividing by zero has no answer. Some calculators print a huge float like 1.6 × 10¹⁶; this one says so plainly instead.
Why does 5! equal 120?
Because 5! is 5 × 4 × 3 × 2 × 1, which is 120. Factorial is defined for whole numbers only — 3.5! has no value here — and 0! is 1 by convention.
Can I use my keyboard?
Yes — click the expression box and type: digits, +, −, ×, ÷, ^, %, ., parentheses and Enter all work. Function names work too: type sin(30°) exactly as you would write it.
Why does 0.1 + 0.2 equal 0.3 here and 0.30000000000000004 elsewhere?
Because the plain arithmetic here uses exact fractions, never floats: 0.1 + 0.2 is exactly 0.3. Most calculators store decimals as binary floats, which cannot represent 0.1 exactly; this one stores the fraction 3/10.
What happens if I divide by zero?
The calculator says "Cannot divide by zero" instead of guessing. Division by zero has no answer, so there is nothing honest to display — fix the divisor and try again.