Step 1: Express the decimal as a fraction

Convert the decimal number $0.000001$ into fractional form. Since there are 6 decimal places, it can be written as $1$ over $1,000,000$ (or $10^6$).

$0.000001 = \frac{1}{1,000,000} = \frac{1}{10^6}$

Step 2: Apply the fractional exponent

Substitute the fraction into the given expression with the power of $1/3$, which represents taking the cube root.

$\left(\frac{1}{10^6}\right)^{1/3} = \frac{1^{(1/3)}}{(10^6)^{(1/3)}}$

Step 3: Simplify the powers

Using the exponent law $(a^m)^n = a^{m \times n}$, multiply the exponent $6$ by $1/3$ to get $2$.

$10^{6 \times \frac{1}{3}} = 10^2 = 100$

Step 4: Convert back to decimal form

Calculate the final value as a decimal fraction: $\frac{1}{100} = 0.01$.

$\frac{1}{100} = 0.01$