What a negative exponent really means, how to rewrite it as a fraction, and how to write very large and very small numbers in scientific notation — the Grade 9 extension of the exponent laws, with worked examples and a quiz.
Grade 8 kept every exponent positive. Grade 9 removes that restriction, and the moment negatives are allowed you can describe numbers as small as the width of an atom just as easily as numbers as large as the distance to the sun.
1Where negative exponents come from
Follow the pattern downwards and each step divides by the base:
- 23 = 8
- 22 = 4 (÷ 2)
- 21 = 2 (÷ 2)
- 20 = 1 (÷ 2)
- 2−1 = ½ (÷ 2 again)
- 2−2 = ¼
A negative exponent does NOT make the answer negative. 2−3 = ⅛ — a positive fraction, not −8. The minus sign means "reciprocal", not "negative".
- 1Flip it: 5−2 = 152.A negative exponent means take the reciprocal.
- 252 = 25.
- 3So 5−2 = 125.The answer is positive.
- 1Subtract the exponents: x5−8 = x−3.The division law still applies.
- 2Rewrite with a positive exponent: x−3 = 1x3.Final answers are normally given without negative exponents.
A neat shortcut: a factor simply moves across the fraction line and its exponent changes sign. So 1a−2 = a2.
2Scientific notation for large numbers
- 1Put the comma after the first digit: 4,5A must be between 1 and 10.
- 2Count the places moved: 6.The comma jumped 6 places to the left.
- 34 500 000 = 4,5 × 106.Moving left gives a positive exponent.
3Scientific notation for small numbers
- 1Move the comma to sit after the 7: 7,2Again A must be between 1 and 10.
- 2The comma moved 4 places to the right.
- 30,00072 = 7,2 × 10−4.Moving right gives a NEGATIVE exponent.
Quick check: numbers bigger than 1 take a positive exponent; numbers smaller than 1 take a negative one.
Practice exercises
Work each one out, then click to reveal the answer.
- 1Write 3−2 as a fraction.
Show answer ▾Hide answer ▴
132 = 19 - 2Write 2−4 as a fraction.
Show answer ▾Hide answer ▴
116 - 3Write 1x5 using a negative exponent.
Show answer ▾Hide answer ▴
x−5 - 4Simplify: y3 ÷ y7 (positive exponents only).
Show answer ▾Hide answer ▴
y−4 = 1y4 - 5Calculate: 10−3
Show answer ▾Hide answer ▴
11000 = 0,001 - 6Write 62 000 000 in scientific notation.
Show answer ▾Hide answer ▴
6,2 × 107 - 7Write 0,0045 in scientific notation.
Show answer ▾Hide answer ▴
4,5 × 10−3 - 8Write 3,1 × 10−2 as an ordinary number.
Show answer ▾Hide answer ▴
0,031 - 9Write 8,04 × 105 as an ordinary number.
Show answer ▾Hide answer ▴
804 000
Quick Quiz
4 quick questions on what you just read. Take it when you feel ready.
Now practise it
Download Grade 9 past papers and worksheets on this topic.