MISSION · Number sense
Exponents and Powers
Repeated multiplication, and how fast it runs away from you
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
An exponent counts how many copies of the base are multiplied together — not how many times you add it, which is the mistake almost everyone makes once.
\[2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\]Multiplied, not multiplied by
\(2^5\) is not \(2 \times 5\). One is 32 and the other is 10, and they part company fast: \(2^{10}\) is 1024 while \(2 \times 10\) is 20.
That gap is the reason exponents matter. Repeated addition grows in a straight line; repeated multiplication bends upward and keeps bending. Folding a piece of paper 42 times would, in principle, reach the moon, and that sentence is only surprising if you are still thinking additively.
The rules, and where they come from
You never have to memorise these if you’re willing to write the multiplication out.
Multiplying powers of the same base — add the exponents. \(2^3 \times 2^4 = (2 \times 2 \times 2)(2 \times 2 \times 2 \times 2) = 2^7\) Three twos next to four twos is seven twos. That’s all the rule says.
Dividing — subtract the exponents. \(\frac{2^5}{2^3} = 2^2\), because three of the five twos cancel against the bottom.
A power of a power — multiply the exponents. \((2^3)^2 = 2^3 \times 2^3 = 2^6\). Two lots of three twos.
Zero and negative exponents
\(2^0 = 1\), which looks like it should be 0 until you follow the pattern down. Each step down divides by 2: \(8, 4, 2, \ldots\) and the next one is 1. The division rule agrees: \(\frac{2^3}{2^3} = 2^0\), and anything divided by itself is 1.
Keep going and the exponents go negative, which means dividing rather than multiplying:
\[2^{-1} = \frac{1}{2} \qquad 2^{-3} = \frac{1}{8}\]A negative exponent never makes a negative number. It makes a fraction.
Squares and cubes
\(5^2\) is called “five squared” because it is the area of a 5 by 5 square. \(5^3\) is “five cubed” — the volume of a 5 by 5 by 5 cube. The names are geometric facts, not decoration, and they are the reason area units are squared and volume units are cubed.
Common mix-ups
- Reading \(3^4\) as 12. It is \(3 \times 3 \times 3 \times 3 = 81\).
- Adding exponents when the bases differ. \(2^3 \times 3^3\) is not \(6^6\). The rule needs the same base.
- Thinking a negative exponent gives a negative answer. It gives a reciprocal.
- Applying an exponent across addition. \((a + b)^2\) is not \(a^2 + b^2\); expand it and you’ll find a middle term.
Check yourself
Work out \(3^4\), then simplify \(\frac{5^6}{5^4}\) and \(10^{-2}\).
\(3^4 = 81\). \(\frac{5^6}{5^4} = 5^2 = 25\). \(10^{-2} = \frac{1}{100} = 0.01\).
LEVEL 3 · REFLECT
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