Solve (1 + x) to the 15th power = 1.0306.

Step-by-step solution

1. 1

   Set up the inverse operation. The unknown sits inside a $15$th power, so raise both sides to the reciprocal exponent: $1+x = 1.0306^{1/15} = \sqrt[15]{1.0306}.$ An odd root has a unique real value, so there is no sign ambiguity here.

2. 2

   Use logarithms rather than repeated roots. $\ln 1.0306 = 0.03014116$. Dividing by $15$: $\tfrac{0.03014116}{15} = 0.00200941$.

3. 3

   Exponentiate back. $1+x = e^{0.00200941} = 1.00201143.$ Note that $e^{t}\approx 1+t$ for small $t$, which is why the exponent $0.00200941$ and the result's fractional part $0.00201143$ look almost identical - they differ only in the fifth significant figure.

4. 4

   Subtract 1 for the rate. $x = 1.00201143 - 1 = 0.00201143 \approx 0.2011\%\ \text{per period}.$

5. 5

   Check by compounding forward. $1.00201143^{15} = 1.030600$, exactly the given factor.

6. 6

   Keep enough digits. Rounding the root to $1.0020$ would give $0.20\%$ and, compounded $15$ times, only $1.03045$ - already off in the fourth decimal of the growth factor. With small rates and many periods, carry at least eight significant figures until the final rounding.