Powers, Square Roots and Cube Roots

An index (or power) is a shorthand for repeated multiplication. In 2⁵, the 2 is the base — the number being multiplied — and the 5 is the index, telling you how many of them there are. So 2⁵ means 2 × 2 × 2 × 2 × 2, which is 32. The mistake almost everyone makes at first is reading it as 2 × 5 = 10. The index is not one of the numbers being multiplied; it counts them. Whenever you are unsure, write the multiplication out in full — it takes ten seconds and it is always right.

Two lists are worth knowing by heart. The square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 — these are n times n. And the cubes 1, 8, 27, 64, 125, 216 — n times n times n. Memorising them is the single highest-value thing in this chapter, because roots, surds, Pythagoras and factorising all lean on recognising them instantly. Spotting that 144 is 12 squared is worth more than any method. One curiosity worth noticing: 64 appears in both lists, because it is 8 squared and also 4 cubed.

A root is a power asked backwards. Where 7 squared equals 49 asks 'what is 7 squared', the square root of 49 asks 'what number squared gives 49?' — and the answer is 7. Same relationship, read the other way. The square root of 144 is 12, and the cube root of 27 is 3 because 3 times 3 times 3 is 27. Watch the symbol: a plain root sign means square root, with the 2 left off by convention, while a small 3 in front means cube root. That matters — the cube root of 64 is 4, but the square root of 64 is 8.

Most numbers are not perfect squares, so the root falls between two whole numbers. To estimate the square root of 50, find the perfect squares either side: 49 and 64. So the root lies between 7 and 8. Since 50 is only just past 49, the root is only just past 7 — about 7.07. The trap is assuming the answer sits midway. Testing 7.5 gives 56.25, which is well above 50, so the root must be much closer to 7 than to 8. Squaring stretches the gaps unevenly, so always test the midpoint rather than guessing.

Two final things. First, negative bases: count the minus signs. Negative 3 squared is negative 3 times negative 3, which is positive 9 — an even index gives a positive result. Negative 3 cubed is negative 27, because an odd index leaves one minus over. But brackets matter enormously: −3² without brackets means only the 3 is squared, giving −9. Second, indices come before multiplying and adding in the order of operations. So 3 × 2³ is 3 × 8 = 24, not 6³, and 5 + 4² is 5 + 16 = 21. The habit that prevents both errors is to work out every power first, then do everything else.