A number times itself

Liri tried to write down the distance to her home star – a number with countless zeros. “That is far too long for me, I keep miscounting!”

Erklärus: “There is a shortcut for exactly that: the power. Astronomers never write out huge distances with all the zeros – they use powers. Learn this, and you can finally write down your own distance from home neatly.”

🔟 What is a power?

A power is made of two parts: the base (the number that is multiplied) and the exponent or index (how many times it is multiplied by itself).

2 3 Base Exponent
Liri: “So 2³ means that I multiply the 2 by itself three times?”
Erklärus: “Exactly. 2³ = 2 × 2 × 2 = 8. The exponent only tells you HOW MANY TIMES the base appears in the multiplication – not that you multiply by the exponent.”
Work out: 3⁴ =
Liri: “With powers I could even write down how much weaker my distress signal gets with distance – that also decreases in steps.”

✖️ Multiplying powers – same base

Erklärus: “What is x³ × x²? Write it out in full, without the shortcut.”
Liri: “x³ is x×x×x, and x² is x×x. Together that is… x×x×x×x×x – so five times x. That is x⁵!”
🧠 Why do you add the exponents?

x³ × x² = (x×x×x) × (x×x) = x×x×x×x×x = x⁵

You simply count how many x’s there are altogether: 3 + 2 = 5.

Rule

aᵐ × aⁿ = aᵐ⁺ⁿ

Only if the base is the same!

Simplify: x⁴ × x³ = x?

➗ Dividing powers – same base

Erklärus: “And x⁷ ÷ x³? When dividing, equal factors cancel each other out.”
Liri: “So of the 7 x’s only 7 minus 3 are left – 4 of them. x⁴!”
Rule

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Simplify: x⁹ ÷ x⁴ = x?

🔁 A power of a power & special cases

Erklärus: “(x²)³ means: take x² and multiply it by itself three times. That is x²×x²×x² – together x⁶.”
Rule

(aᵐ)ⁿ = aᵐ×ⁿ

Liri: “And what is 2⁰ actually? Multiplying zero times – is that 0 then?”
Erklärus: “A good suspicion, but wrong. Look at this pattern – each time you divide by 2:”
2³ = 8
2² = 4
2¹ = 2
2⁰ = 1
2⁻¹ = ½
2⁻² = ¼
Erklärus: “Each step is divided by 2: 8→4→2→1. And after that it simply carries on: 1 divided by 2 is a half. That is how the rule for negative exponents comes about – no trick, just a pattern that you keep thinking through.”
Special cases

a⁰ = 1  (a ≠ 0)

a⁻ⁿ = 1 ÷ aⁿ

Work out: 7⁰ =
🌌 Powers in the universe

The number of stars in the observable universe is estimated at about 10²⁴ – a 1 followed by 24 zeros.
A single atom is about 10⁻¹⁰ metres across – negative exponents describe unimaginably small sizes.
The speed of light is about 3 × 10⁸ m/s.

❌ Common mistakes

2³ is not 2×3=6! The exponent is not a second multiplication, but counts how often the base itself appears: 2³=2×2×2=8.

Also, x³ + x² can not be combined into x⁵! The laws of powers only apply to multiplying and dividing, not to adding.

📝 Exercise 1
Simplify: a⁵ × a³ = a?
📝 Exercise 2
Simplify: (x³)² = x?
📝 Exercise 3
Work out: 2⁻² = 1/?

Big numbers, written short

Liri now wrote her distance from home as a power – short, clear, and immediately understandable. At last she could compare it with other distances without getting lost in endless zeros.

Liri: “Now I can compare my distance home with the distances of other stars – power against power. That suddenly makes it something I can grasp, no longer just an unimaginable number.”
Erklärus: “That is exactly how real astronomers navigate: they don’t compare the written-out numbers, but the exponents. Anyone who masters powers can place even gigantic distances – even their own way home.”
“Power: base to the power of the exponent. aᵐ×aⁿ=aᵐ⁺ⁿ, aᵐ÷aⁿ=aᵐ⁻ⁿ, (aᵐ)ⁿ=aᵐ×ⁿ. a⁰=1, a⁻ⁿ=1÷aⁿ. Only usable with the same base, never when adding.”
What you learned today
aᵐ×aⁿ=aᵐ⁺ⁿ
Same base: add, subtract or multiply the exponents
🌟

Chapter 45 mastered!

Liri can now write huge and tiny numbers down short and clear.

Erklärus’s notebook – page 45:
“The laws of powers only apply with the same base. The exponent counts how often the base is multiplied – it is not a second factor in the multiplication.”
← Back to overview 45 – Powers