Exponent: definition and properties

Power aⁿ equals the product of a number a on itself n times.

aⁿ = a * a * a… a (n once)

In this case, a is the foundation, and n – exponent.

examples:

• 3¹ = 3
• 3² = 3 x 3 = 9
• 3³ = 3 x 3 x 3 = 27
• 3⁴ = 3 x 3 x 3 x 3 = 81
• 3⁵ = 3 × 3 × 3 × 3 × 3 = 243

Pronunciation:

• Number a² should be pronounced like “a squared”. For example, 4² is “four squared”.
• Number a³ pronounced as “a cubed”. For example, 4³ is “four cubed”.
• In all other cases aⁿ it says like “a to the nth power”. For example, 4⁶ is “four to the sixth power”.

Rules for operations with exponents

#one. Multiplication of powers (same bases)

aⁿ ⋅ a^m = A^n+m

Example: 2² ⋅2³ = 2^{2 + 3} = 2⁵ = 32

#2. Product degree

(a ⋅ b)ⁿ  = Aⁿ ⋅ bⁿ

Example: (2 ⋅ 3)⁴ = 2⁴ ⋅3⁴ = 1296

#3. Division of powers (same bases)

aⁿ / at^m = A^nm

Example: 2⁵ ⋅2³ = 2^{5 – 3} = 2² = 4

#four. Degree of quotient

(a / b)ⁿ  = Aⁿ /bⁿ

Example: (12 / 4)³ = 12³ / 4³ = 27

#5. Exponentiation

(aⁿ)^m = A^{n · m}

Example: (5²)³ = 5^{2 · 3} = 3125

#6. Power raised to a power

a^nm = A^(nm)

Example: 2⁴² = 2⁽⁴²⁾ = 2^{(4 · 4)} = 2⁽¹⁶⁾ = 65536

#7. Extracting a degree from a number to a degree

^m√(aⁿ) = the ^n/m

Example: ³√(2⁶) = 2^6/3 = 2² = 2⋅2 = 4

#eight. Raising to a negative power

b^-n = 1 / bⁿ

Example: 2^-4 = 1 / 2⁴ = 1 / (2⋅2⋅2⋅2) = 1/16 = 0,0625

# 9. Number to the power of zero

a⁰ = 1

Example: 10⁰ = 1

#ten. Raising zero to a power

0ⁿ = 0, for n>0

Example: 0⁷ = 0

#eleven. Number to the first power

a¹ = a

Example: 15¹ = 15

#12. Unit in degree (any)

1ⁿ = 1

Example: 1²⁰ = 1

#13. Minus one to the power

(-1)ⁿ = 1 if n is an even number

(-1)ⁿ = -1 if n is an odd number

Example: (-1)⁶ = 1

#fourteen. Raising a number to a fractional power (the numerator is one)

a^{1 / n} = ⁿ√a

Example: 27^1/3 = ³√27 = 3