Negative Indices

Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.

Progress: 0 / 9 correct

Keep going!

Level 1 · Warm-up

1 Work out 3⁻¹ + 6⁻¹.

Show worked solution

3⁻¹ = 1/3 = 2/6 and 6⁻¹ = 1/6, so the sum is 3/6 = \(\tfrac{1}{2}\). −9 treats the indices as negative numbers; 1/18 multiplies instead of adding; 1/9 adds the bases first.

2 Write 0.0001 as a power of 10, and then as a power of 100.

0.0001 = 10 to the power e.g. 3 0.0001 = 100 to the power e.g. 5

Show worked solution

0.0001 = 1/10 000 = 10⁻⁴, so the power of 10 is −4. Since 100 = 10², 100⁻² = 10⁻⁴, so the power of 100 is −2.

3 Find n if 2ⁿ = 4 ÷ 32.

Show worked solution

4 = 2² and 32 = 2⁵, so 4 ÷ 32 = 2²⁻⁵ = 2⁻³ and n = −3. Check: 2⁻³ = 1/8, and 4/32 = 1/8 ✓.

Level 2 · Core

4 Simplify (2a⁻³)⁻², writing the answer with positive indices.

Show worked solution

2⁻² = 1/4 and (a⁻³)⁻² = a⁶, so the answer is \(\dfrac{a^{6}}{4}\). 4a⁶ forgets that 2⁻² is 1/4; −4a⁶ makes 2⁻² negative; 1/(4a⁶) gets −3 × −2 wrong — it is +6.

5 Simplify 15m⁻⁴n² ÷ 3m²n⁻³. The answer has the form \(\dfrac{a\,n^{b}}{m^{c}}\). Find a, b and c.

a e.g. 2 b e.g. 3 c e.g. 4

Show worked solution

15 ÷ 3 = 5; for m, −4 − 2 = −6; for n, 2 − (−3) = 5. So 5m⁻⁶n⁵ = 5n⁵/m⁶: a = 5, b = 5, c = 6.

6 Work out \(\left(\tfrac{1}{2}\right)^{-4}\) + 4⁻¹ × 4³.

Show worked solution

Flip 1/2 to 2: 2⁴ = 16. 4⁻¹ × 4³ = 4² = 16. Total: 16 + 16 = 32.

Level 3 · Challenge

7 Three statements:
1 (x⁻²y)³ ÷ x⁻⁸ = x²y³
2 2⁻¹ + 2⁻¹ = 2⁻²
3 (5a)⁻¹ × 10a³ = 2a²
Which statements are correct?

Show worked solution

(x⁻²y)³ = x⁻⁶y³, and −6 − (−8) = 2, giving x²y³ (1 ✓). 2⁻¹ + 2⁻¹ = 1/2 + 1/2 = 1 = 2⁰, not 2⁻² = 1/4 (2 ✗). (5a)⁻¹ = 1/(5a), and 10a³ ÷ 5a = 2a² (3 ✓). 1 and 3.

8 Work out (2 × 10⁻³)⁻² and write the answer in scientific notation.

front number × 10 e.g. 1.6 power e.g. 3

Show worked solution

2⁻² = 1/4 = 0.25 and (10⁻³)⁻² = 10⁶, so the answer is 0.25 × 10⁶ = 2.5 × 10⁵.

9 Find n if 0.25 × 2⁻³ ÷ 16⁻² = 2ⁿ.

Show worked solution

0.25 = 1/4 = 2⁻² and 16⁻² = (2⁴)⁻² = 2⁻⁸. The index is −2 + (−3) − (−8) = 3, so n = 3.