Powers and roots

A power is a short way of writing repeated multiplication.

$2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$

The square root goes the other way: $\sqrt{49} = 7$ because $7^2 = 49$.

Why it is worth knowing: powers make it possible to write very large and very small numbers briefly.

A large language model has roughly $10^{11}$ parameters and is trained on roughly $10^{13}$ tokens. Writing those numbers out would be unmanageable.

The laws of exponents — all of them follow from writing the multiplication out:

The row about $a^0 = 1$ surprises many people. Look at the pattern: $2^3 = 8$, $2^2 = 4$, $2^1 = 2$, $2^0 = ?$ — every step down halves, so the next one is 1. And then $2^{-1} = 0.5$.

Two common errors:

The second is one of the most common errors in all school mathematics. Check it with numbers: $(1+2)^2 = 9$, but $1^2 + 2^2 = 5$.

import math

print(2 ** 5, math.sqrt(49), 49 ** 0.5)          # 32 7.0 7.0

# The laws of exponents, checked numerically
assert 2**3 * 2**4 == 2**7
assert 2**5 / 2**2 == 2**3
assert (2**3) ** 2 == 2**6
assert 7**0 == 1
assert 2**-3 == 1 / 8
assert abs(9 ** 0.5 - math.sqrt(9)) < 1e-12
print("all the laws of exponents hold")

# The pattern that explains a⁰ = 1
for e in range(3, -4, -1):
    print(f"2^{e:>2} = {2.0 ** e}")
# 2^ 3 = 8.0
# 2^ 0 = 1.0
# 2^-3 = 0.125

# The classic error — check it with numbers
a, b = 1, 2
print((a + b) ** 2, a**2 + b**2)                 # 9 5  ← not the same thing!
print((a + b) ** 2, a**2 + 2*a*b + b**2)         # 9 9  ← right

# Large and small numbers
print(f"{10**11:,} parameters")                  # 100,000,000,000
print(f"{1.75e12:,.0f} tokens")                  # 1,750,000,000,000
print(2 ** 10, 2 ** 20, 2 ** 30)                 # 1024 1048576 1073741824
#      ↑ kilo      ↑ mega        ↑ giga (in the computers' powers of two)