Let's use this algorithm to raise 3 to the power 7.
<UL>
  <LI>P1: raise 3 to the power 2 (choose I=2, no particular reason).
      <UL>
        <LI>Solve directly: S1=9.
      </UL>
  <LI>P2: raise 3 to the power 5 (7-2=5).
      <UL>
        <LI>P2.1: raise 3 to the power 1. (choose I=1)
            <UL>
              <LI>Solve directly: S2.1=3.
            </UL>
        <LI>P2.2: raise 3 to the power 4 (5-1=4).
            <UL>
              <LI>P2.2.1: raise 3 to the power 2.
                  <UL>
                    <LI>Solve directly: S2.2.1=9.
                  </UL>
              <LI>P2.2.2: raise 3 to the power 2.
                  <UL>
                    <LI>Solve directly: S2.2.2=9.
                  </UL>
              <LI>Combine: S2.2 = S2.2.1 * S2.2.2 = 9 * 9 = 81.
            </UL>
        <LI>Combine: S2 = S2.1 * S2.2 = 3 * 81 = 243.
      </UL>
  <LI>Combine: S = S1 * S2 = 9 * 243 = 2147.
</UL>