Aaron Walcott, TT
Indices and Powers
Indices are the powers that a base is raised to, derived from repeated representation in a
multiplication or division.
If we began with ‘a’ and multiplied it again by itself,
thus a x a, I’d have a2
a x a = a2 (1) (pronounced “a to the power of 2,” or “a squared.”)
a x a x a = a3 (2) (pronounced “a to the power of 3,” or “a cubed.”)
a x a x a x a = a4 (3) (pronounced “a to the power of 4”) etc
Here, the base is ‘a’ (think of the base as a literal description, because it sits on the line) and ‘2’
as the power (above the line) that ‘a’ is “raised to” (the index). This is older language, but it
helps when visualizing these problems in their alternate forms.
If we had to multiply line (1) by line (2)
a x a = a2 x a x a x a = a3 =
a2 x a3 = a x a x a x a x a = a5 (we can get this just by counting how many times a is repeated) or
we can add: index 2 plus index 3 gives index 5
If we had to multiply line (2) by line (3)
a x a x a = a3 x a x a x a x a = a4 =
a3 x a4 = a x a x a x a x a x a x a = a7 (we can again get this just by counting how many times a is
repeated) or
we can add: index 3 plus index 4 gives index 7
and so, we arrive at the Product Rule cited below.
All these rules can be derived from first principles or internal visualization, once this first rule is
established mentally. We can extend this rule to division and use subtraction to establish the
Quotient Rule of Indices (you may see “exponents” in many textbooks) and so on and so forth.
1
, Aaron Walcott, TT
Law of Exponents
m n m +n
Product Rule a × a =a
m n m−n
Quotient Rule a ÷ a =a
Power of a Power Rule ( a m ¿n=a mn
Power of a Product Rule ( ab ¿m=am bm
()
m m
a a
Power of a Quotient Rule = m
b b
0
Zero Exponent Rule a =1
−m 1
Negative Exponent Rule a =
am
m
Fractional Exponent Rule a n =√ a
n m
Rules for Radicals
1
Normal Root √n x=x n
m
Power Inside Root √n x m=x n
Multiplication √n x ⋅ y =√n x ⋅ √n y
Division
√n x √n x
=
y √n y
Addition & Subtraction of powers inside root
√ an +b n ≠ a ± b
Addition & Subtraction inside root
√n x ± y ≠ √n x ± √n y
2
Indices and Powers
Indices are the powers that a base is raised to, derived from repeated representation in a
multiplication or division.
If we began with ‘a’ and multiplied it again by itself,
thus a x a, I’d have a2
a x a = a2 (1) (pronounced “a to the power of 2,” or “a squared.”)
a x a x a = a3 (2) (pronounced “a to the power of 3,” or “a cubed.”)
a x a x a x a = a4 (3) (pronounced “a to the power of 4”) etc
Here, the base is ‘a’ (think of the base as a literal description, because it sits on the line) and ‘2’
as the power (above the line) that ‘a’ is “raised to” (the index). This is older language, but it
helps when visualizing these problems in their alternate forms.
If we had to multiply line (1) by line (2)
a x a = a2 x a x a x a = a3 =
a2 x a3 = a x a x a x a x a = a5 (we can get this just by counting how many times a is repeated) or
we can add: index 2 plus index 3 gives index 5
If we had to multiply line (2) by line (3)
a x a x a = a3 x a x a x a x a = a4 =
a3 x a4 = a x a x a x a x a x a x a = a7 (we can again get this just by counting how many times a is
repeated) or
we can add: index 3 plus index 4 gives index 7
and so, we arrive at the Product Rule cited below.
All these rules can be derived from first principles or internal visualization, once this first rule is
established mentally. We can extend this rule to division and use subtraction to establish the
Quotient Rule of Indices (you may see “exponents” in many textbooks) and so on and so forth.
1
, Aaron Walcott, TT
Law of Exponents
m n m +n
Product Rule a × a =a
m n m−n
Quotient Rule a ÷ a =a
Power of a Power Rule ( a m ¿n=a mn
Power of a Product Rule ( ab ¿m=am bm
()
m m
a a
Power of a Quotient Rule = m
b b
0
Zero Exponent Rule a =1
−m 1
Negative Exponent Rule a =
am
m
Fractional Exponent Rule a n =√ a
n m
Rules for Radicals
1
Normal Root √n x=x n
m
Power Inside Root √n x m=x n
Multiplication √n x ⋅ y =√n x ⋅ √n y
Division
√n x √n x
=
y √n y
Addition & Subtraction of powers inside root
√ an +b n ≠ a ± b
Addition & Subtraction inside root
√n x ± y ≠ √n x ± √n y
2
