A minus sign in the power flips it downstairs: Think of powers as repeated multiplication, then count backwards. Each step down divides by the base:
8 = 2³, 4 = 2², 2 = 2¹, 1 = 2⁰, ½ = 2⁻¹, ¼ = 2⁻².
So a negative exponent just means a reciprocal:
x⁻ⁿ = 1/xⁿ. The minus does NOT make the answer negative — it sends the power to the bottom of a fraction.
Why this matters in AI: decay and 'per unit' rates (like a population that halves, or a price that drops by a factor each year) are written with negative powers, e.g. 2⁻ᵗ.
IB-style question — evaluate a negative power
A bacterial sample is modelled so that its mass (in mg) is M = 5 × 2⁻ᵏ, where k is the number of cooling cycles.
Find the mass after 3 cooling cycles.
Step by step
- Substitute k = 3. The negative power means a reciprocal.
- Evaluate 2³ = 8, then multiply.
Final answer
M = 5/8 = 0.625 mg. (The negative power shrank the mass, as expected for cooling/decay.)
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The denominator of the power is the root: A power of 1/n undoes an n-th power, so it is the n-th root:
x1/2 = √x (square root), x1/3 = ³√x (cube root).
Why? Because (x1/2)² = x1/2 × 2 = x¹ = x — squaring it gives back x, which is exactly what 'square root' means.
For a general fraction m/n, the top is the power and the bottom is the root:
xm/n = ⁿ√(xᵐ) = (ⁿ√x)ᵐ.
Do whichever order is easier — usually take the root first to keep numbers small.
IB-style question — evaluate a fractional power
Find the exact value of 272/3.
Step by step
- Bottom is 3 → cube root; top is 2 → square it. Take the root first (smaller numbers).
- The cube root of 27 is 3.
- Square it.
Final answer
272/3 = 9.
IB-style question — combine a fraction and a minus
The volume V (in m³) of a storage tank scales with a design parameter d by V = 16−3/4 × d.
Find the constant 16−3/4 as an exact fraction.
Step by step
- The minus → reciprocal; the bottom 4 → 4th root; the top 3 → cube it.
- The 4th root of 16 is 2 (since 2⁴ = 16); cube it.
Final answer
16−3/4 = 1/8, so V = d/8.