What are the various laws of exponent? Explain with examples.
Answers
Answer:
What is an exponent
The base a raised to the power of n is equal to the multiplication of a, n times:
a n = a × a × ... × a
n times
a is the base and n is the exponent.
Examples
31 = 3
32 = 3 × 3 = 9
33 = 3 × 3 × 3 = 27
34 = 3 × 3 × 3 × 3 = 81
35 = 3 × 3 × 3 × 3 × 3 = 243
Exponents rules and properties
Rule name Rule
Product rules a n ⋅ a m = a n+m
a n ⋅ b n = (a ⋅ b) n
Quotient rules a n / a m = a n-m
a n / b n = (a / b) n
Power rules (bn)m = bn⋅m
bnm = b(nm)
m√(bn) = b n/m
b1/n = n√b
Negative exponents b-n = 1 / bn
Zero rules b0 = 1
0n = 0 , for n>0
One rules b1 = b
1n = 1
Minus one rule
Derivative rule (xn)' = n⋅x n-1
Integral rule ∫ xndx = xn+1/(n+1)+C
Exponents product rules
Product rule with same base
an ⋅ am = an+m
Example:
23 ⋅ 24 = 23+4 = 27 = 2⋅2⋅2⋅2⋅2⋅2⋅2 = 128
Product rule with same exponent
an ⋅ bn = (a ⋅ b)n
Example:
32 ⋅ 42 = (3⋅4)2 = 122 = 12⋅12 = 144
See: Multplying exponents
Exponents quotient rules
Quotient rule with same base
an / am = an-m
Example:
25 / 23 = 25-3 = 22 = 2⋅2 = 4
Quotient rule with same exponent
an / bn = (a / b)n
Example:
43 / 23 = (4/2)3 = 23 = 2⋅2⋅2 = 8
See: Dividing exponents
Exponents power rules
Power rule I
(an) m = a n⋅m
Example:
(23)2 = 23⋅2 = 26 = 2⋅2⋅2⋅2⋅2⋅2 = 64
Power rule II
a nm = a (nm)
Example:
232 = 2(32) = 2(3⋅3) = 29 = 2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅2 = 512
Power rule with radicals
m√(a n) = a n/m
Example:
2√(26) = 26/2 = 23 = 2⋅2⋅2 = 8
Negative exponents rule
b-n = 1 / bn
Example:
2-3 = 1/23 = 1/(2⋅2⋅2) = 1/8 = 0.125
Answer:
Laws of Exponents. When multiplying like bases, keep the base the same and add the exponents. When raising a base with a power to another power, keep the base the same and multiply the exponents. When dividing like bases, keep the base the same and subtract the denominator exponent from the numerator exponent.