8 to the 26th power

Question

what is 8 to the twenty-sixth power

Answer 1

To calculate this accurately we can use the fact that 8=10-2.

(10-2)²⁶ has the following successive coefficients:

1, 26, 325, 2600, 14950, 65780, 230230, 657800, 1562275, 3124550, 5311735, 7726160, 9657700, 10400600, 9657700, 7726160, 5311735, 3124550, 1562275, 657800, 230230, 65780, 14950, 2600, 325, 26, 1.

We also need successive powers of 2:

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864.

Powers of 10 are easy: 10²⁶=100,000,000,000,000,000,000,000,000 (1 followed by 26 zeroes). This is the first term. Most calculators display about 9 decimal places and a power of 10. We need abut three times this accuracy. The series of terms alternates between positive and negative terms. The odd terms are positive and the even terms are negative. It is best to sum all the positives separate from the negatives and carry out the subtraction later.

All we need now is patience, care and a calculator.

10²⁶-26(10²⁵)(2)+325(10²⁴)(2²)-

The tables summarise the work involved. 8²⁶=302,231,454,903,657,293,676,544.