Convert between standard and scientific notation, and multiply or divide numbers written in it — the exponent tracks how many places the decimal moved, and losing count of that is where marks go. Every question comes with a written walkthrough of the conversion.
Moving the decimal one place too many or too few changes the exponent by exactly that much — always recount by moving the decimal one digit at a time rather than estimating.
Multiplying two coefficients can produce a result of 10 or greater, which no longer has exactly one nonzero digit before the decimal — that extra factor of 10 has to be folded back into the exponent.
A small number like 0.00042 needs a negative exponent, and a large number like 42,000 needs a positive one — mixing up which direction the decimal moved flips the sign.
Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.
Write 0.0056 in scientific notation.
A — 5.6 × 10⁻³. Move the decimal point right until exactly one nonzero digit remains before it: 0.0056 becomes 5.6 after moving three places, so the exponent is −3, giving 5.6 × 10⁻³. 5.6 × 10³ (B) has the correct coefficient but the wrong sign on the exponent — a number smaller than 1 always needs a negative exponent, not positive.
Compute: (3 × 10⁴)(2 × 10⁻⁷)
A — 6 × 10⁻³. Multiply the coefficients: 3 × 2 = 6. Add the exponents: 4 + (−7) = −3. The result is already normalized, giving 6 × 10⁻³. 6 × 10¹¹ (B) comes from treating the −7 as +7 when adding the exponents — a sign-tracking slip on the negative exponent that gives 4 + 7 = 11 instead of 4 + (−7) = −3.
Compute and write in scientific notation: (4.5 × 10⁶) ÷ (9 × 10⁻²)
A — 5 × 10⁷. Divide the coefficients: 4.5 ÷ 9 = 0.5. Subtract the exponents: 6 − (−2) = 8, giving 0.5 × 10⁸. That coefficient isn't normalized — 0.5 has no nonzero digit before the decimal — so shift it: 0.5 × 10⁸ = 5 × 10⁷. 0.5 × 10⁸ (C) is mathematically equal but stops one step early, before the re-normalization that scientific notation requires.
Scientific notation rewards precision over speed at first — miscounting a decimal place by one is invisible until the exponent comes out wrong.
Move the decimal a single digit at a time and count out loud or on paper rather than estimating the shift. Untimed practice is where that precision gets built.
Move to timed sessions once counting is automatic and re-normalizing is a reflex. Scientific notation is one of the quickest subtopics once the habit is set.
Pair scientific notation with exponents in a mock — the underlying exponent rules are identical.
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