Solve (1 + x) to the 20th power = 1.1207.

Step-by-step solution

1. 1

   Recognise the structure. This is a compound-growth equation: $1.1207$ is the total growth factor accumulated over $20$ equal periods, and $x$ is the rate per period. Because the exponent is even, the real $20$th root is taken as the positive one (a negative base would make $1+x<0$, meaningless as a growth factor).

2. 2

   Undo the power with a root. Raising both sides to the $\tfrac{1}{20}$ power gives $1+x = 1.1207^{1/20} = \sqrt[20]{1.1207}.$

3. 3

   Evaluate the root through logarithms. $\ln 1.1207 = 0.11395349$, so $\tfrac{1}{20}\ln 1.1207 = 0.00569767$ and $1+x = e^{0.00569767} = 1.00571394.$ Using logs avoids the rounding drift that repeated square roots would introduce.

4. 4

   Subtract 1 to isolate the rate. $x = 1.00571394 - 1 = 0.00571394 \approx 0.5714\%\ \text{per period}.$

5. 5

   Verify by compounding forward. $1.00571394^{20} = 1.120700$, which reproduces the right-hand side to six decimals, so the root is correct.

6. 6

   Note why the linear estimate is close but not equal. Dividing the total growth evenly would suggest $\tfrac{0.1207}{20}=0.60\%$; the true rate $0.5714\%$ is smaller because compounding does part of the work. Over $20$ periods the gap is about $5\%$ of the rate.