Answer Precision Rescue

Sig figs, scientific notation, and online homework boxes

For the moment before submitting

If the physics is right, do not let formatting make it feel wrong.

There are two different jobs: written physics reporting and typing into an online answer box. Written work should show significant figures cleanly. Online boxes often grade with a tolerance, so the safest entry can be one extra guard digit unless the question gives an exact rounding instruction.

Specific instruction winsIf it says "2 significant figures" or "nearest tenth," obey that.

No instruction?Use 3 significant figures or one guard digit. Do not round every step.

Trailing zeros?Use scientific/E notation if accepted: 6.9e2 means 6.9 x 10^2.

The answer-box rule that matters most

Canvas-style systems usually grade a number, not your reasoning. If the box is numeric-only, it may not care whether your written final answer has perfect sig figs. It may just check whether your typed number is close enough to the expected value.

Best default when no rounding instruction is shown: solve with guard digits, then type 3 significant figures or one extra digit past the rounded answer. Example: if written work would be 6.9 x 10^2 N, the answer box may accept 690, 6.9e2, or a less-rounded value such as 693.7.

If the problem explicitly says 2 significant figures: report 2 significant figures in written work. For an online numeric box, 6.9e2 is often clearer than 690 because the two significant figures are visible.

Do not type units into a numeric-only box unless the box or directions asks for units. A correct value can be rejected if the grader expects only the number.

Written answer vs online answer

Written workShow the teacher you know precision: 6.9 x 10^2 N, not just 690 N, when two significant figures matter.

Online boxType what the grader can parse: 690, 6.9e2, or the guard-digit value if no rounding rule is stated.

Guard digitsKeep extra digits while solving, like 693.7. Round only the final answer, not every line.

E notation2.9e3 means 2.9 x 10^3. It is plain-text scientific notation.

If it marks a correct-looking answer wrong

Re-read the line above the answer box. It may say "nearest tenth," "integer," "2 sig figs," or "do not include units."
Check units. If you calculated centimeters but the box wants meters, the number changes by 100.
Check whether it wants magnitude only. A force of -220 N might need 220 plus "opposite motion" in written work.
If the system allows another attempt and no rounding instruction was given, try one more guard digit before assuming the physics is wrong.
If the answer ends in zeros, try E notation: 9.4e2 instead of only 940, if the box accepts it.

Worksheet-style examples

These match the kind of Chapter 4 and 5 answers that create the most stress: values that round to trailing zeros or values where the calculator shows more digits than the final answer should show.

Calculator value	Teacher-style final	Online box strategy	Why
`693.7 N`	`6.9 x 10^2 N`	`690` or `6.9e2`; if rejected and no strict sig-fig instruction, try `693.7`	Two significant figures keep 6 and 9. The next digit is 3, so no round up.
`2896 N`	`2.9 x 10^3 N`	`2900` or `2.9e3`; if rejected, try `2896`	Scientific notation shows that `2900` has two trusted digits, 2 and 9.
`944`	`9.4 x 10^2 = 940`	`940` or `9.4e2`	`9.44 x 10^2` to two sig figs becomes `9.4 x 10^2`.
`0.2449`	`0.24`	`0.24`; if tolerance is strict and no instruction, try `0.245`	Leading zeros are not significant. The first meaningful digit is 2.
`8.353`	`8.4` for 2 sig figs	`8.4`; if no instruction, `8.35` is safer	The third digit is 5, so `8.3` rounds to `8.4`.
`18.52`	`18.5` for 3 sig figs	`18.5`; if no instruction, `18.52` may be accepted	For 3 sig figs, keep 1, 8, and 5. The next digit is 2.

Scientific notation without panic

Scientific notation separates size from precision. The power of ten tells the size. The coefficient tells the significant figures.

Move decimal right2.9 x 10^3
= 2.9 x 1000
= 2900

Move decimal left3.2 x 10^-3
= 0.0032

Answer-box form6.9e2
= 6.9 x 10^2
= 690

Why trailing zeros feel confusing: plain 940 could mean two, three, or maybe even more significant figures depending on class convention. 9.4 x 10^2 removes the ambiguity because the coefficient 9.4 has exactly two significant figures.

E notation, x10 notation, and negative powers

e is calculator shorthand for "times ten to the power." It is not the letter variable e from algebra here. In physics answer boxes, 6.67e-11 means the same thing as 6.67 x 10^-11.

Read it out loud: 6.67e-11 = "six point six seven times ten to the negative eleventh." The negative belongs to the exponent, so the decimal moves left 11 places.

Teacher notation	Calculator / Canvas notation	Decimal meaning	How to think
`6.67 x 10^-11`	`6.67e-11`	`0.0000000000667`	Move the decimal left 11 places.
`1.07 x 10^-8`	`1.07e-8`	`0.0000000107`	Tiny force from universal gravitation.
`9.4 x 10^2`	`9.4e2`	`940`	Move the decimal right 2 places.
`2.9 x 10^3`	`2.9e3`	`2900`	Move the decimal right 3 places.

Common notation mistakes

Wrong: reading 6.67e-11 as 6.67 x -11. The -11 is an exponent on 10, not a number being multiplied directly.
Wrong: typing 10x-11. Use x 10^-11 in written work or e-11 in a plain answer box.
Right: 6.67e-11, 6.67E-11, and 6.67 x 10^-11 all describe the same value.
Calculator habit: use the EE or EXP key if the calculator has one. For 6.67 x 10^-11, type 6.67 then EE/EXP then -11. Do not type 6.67 x 10 x -11.

Practice 14.2e3 means 4.2 x 10^3 = 4200.

Practice 24.2e-3 means 4.2 x 10^-3 = 0.0042.

Practice 37.8 x 10^2 can be typed as 7.8e2.

Practice 43.1 x 10^-5 can be typed as 3.1e-5.

Calculator buttons: EE, EXP, x10^x, and negative exponents

Most scientific calculators have a shortcut key for scientific notation. It may be labeled EE, EXP, or ×10^x. That key already means "times ten to the power," so you do not press the multiplication key or type another 10 after it.

Goal	What to press	What it means	Common mistake
Enter `6.67 × 10^-11`	`6.67` EE/EXP `-11`	`6.67e-11`	Do not type `6.67 × 10 × -11`.
Enter `2.9 × 10^3`	`2.9` EE/EXP `3`	`2.9e3 = 2900`	Do not type `2.9 × 10 × 3`.
Square a scientific-notation value	Use parentheses: `(2.0e3)^2`	The whole value is squared.	`2.0e3^2` can be interpreted wrong on some calculators.
Divide by `r²` in gravitation	`(6.67e-11)(m1)(m2)/(r^2)`	The exponent stays attached to G.	Forgetting parentheses around the denominator when it has more than one factor.

What the calculator display means

1.07E-8 means 1.07 × 10^-8. Capital E and lowercase e mean the same thing.
-1.07E-8 means the whole number is negative: -(1.07 × 10^-8). The sign in front is separate from the exponent.
1.07E+8 means a large number: 107,000,000. Positive exponent moves right.
1.07E-8 means a tiny number: 0.0000000107. Negative exponent moves left.

Chapter 4 gravitation habit: type the constant as 6.67e-11. Then multiply by both masses and divide by distance squared: (6.67e-11)(m1)(m2)/(r^2). If the answer is tiny, that is normal; gravity between ordinary objects is extremely small.