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  1. 020Without paper or a calculator: what is 9 to the power 6?Mental mathsCoreNomuraTokyo · 2025

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    531,441. Split the power: 9 to the 6th is 9 cubed, squared, and 9 cubed is 729. Square 729 as (730 minus 1) squared: 730 squared is 532,900, less twice 730, which is 1,460, plus 1, gives 531,441. Check it: an even power of 9 ends in 1, the digits add to 18, and 0.9 to the 6th is about 0.53, so the size is right.

    How do you break a big power into steps you can hold in your head?

    Think of carrying a heavy suitcase up six floors: you rest on the landings. Split the exponent into pieces you know, 6 = 3 x 2, so 9 to the 6th is 9 cubed, squared. Most people know 9 squared is 81 and 9 cubed is 729. That turns the question into one square, 729 times 729, which is still awkward until you move it next to a round number.

    Why square 730 instead of 729?

    Because 730 squared is easy: 73 squared is 5,329, so 730 squared is 532,900. The identity (a minus 1) squared equals a squared minus 2a plus 1 lets you square any number just below a round one with two easy corrections. Twice 730 is 1,460. So 532,900 less 1,460 is 531,440, and adding 1 gives 531,441. Each step is small enough to say out loud while you do it.

    The relationship
    96=(93)2=7292=(730−1)2=532,900−1,460+1=531,4419^{6} = (9^{3})^{2} = 729^{2} = (730-1)^{2} = 532{,}900 - 1{,}460 + 1 = 531{,}441
    9^3729, a cube worth knowing by heart
    (a-1)^2a^2 - 2a + 1, here with a = 730
    What it says in wordsTurn the power into one square, then square the nearby round number and correct for the gap.
    Turn 9 to the 6th into one square near a round numberROUTE 1: cube, then square near a round number9 x 9 x 9729729 = 730 - 1square it730² = 532,900- 1,460 + 19 to the 6th531,441(a - 1)² = a² - 2a + 1: 730² = 532,900, 2 x 730 = 1,460, 532,900 - 1,460 + 1 = 531,441ROUTE 2: 81 cubed81 x 816,5616,561 x 80 + 6,561524,880 + 6,56181 cubed531,441Checks: even powers of 9 end in 1. Digits add to 18, divisible by 9. 0.9 to the 6th is about 0.53.
    Cubing 9 gives 729, and squaring it as (730 minus 1) squared gives 532,900 less 1,460 plus 1, which is 531,441. Cubing 81 reaches the same answer, and three quick checks confirm it ends in 1, is divisible by 9 and is about 530 thousand.

    How do you check it before you say it?

    Three checks take seconds. The last digit of an even power of 9 is always 1, the digit sum of any power of 9 is divisible by 9, and 9 to the 6th is 0.9 to the 6th times a million. Here the digits 5, 3, 1, 4, 4, 1 add to 18, and 0.9 to the 6th is about 0.53, so the answer must be about 530 thousand. That last check catches the common slip of being off by a factor of ten. A second route also helps: 81 squared is 6,561, and 6,561 times 81 is 524,880 plus 6,561, again 531,441.

    Where candidates lose it

    The common loss is multiplying 9 by itself six times in sequence and losing the running total around 59,049. Each step adds a chance to slip, and the interviewer watches you stall.

    The second loss is saying a number without a check. An answer like 538,461 sounds plausible; the ends-in-1 and digit-sum checks catch it before it leaves your mouth.

    What the interviewer asks next

    • What is 99 squared, and 999 squared?
    • Roughly, what is 1.09 to the power 6, and what does it mean as compound growth?
    • What is 3 to the power 12?

    Asked at Nomura, Generalist, Tokyo, 2025 (Wall Street Oasis): I think one of the math questions was 9 to the power of 6.