make a list of all laws of exponent
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The laws of Exponents are as follows :
Multiplying powers with the same base
Example : x2 × x3, 23 × 25, (-3)2 × (-3)4
In multiplication of exponents if the bases are same then we need to add the exponents.
NOTE - (i) Exponents can be added only when the bases are same.
(ii) Exponents cannot be added if the bases are not same like.
2.Dividing powers with the same base
Example : 35 ÷ 31, 22 ÷ 21, 5(2) ÷ 53
In division if the bases are same then we need to subtract the exponents.
NOTE -
Where m and n are whole numbers and m n;
am ÷ an = am/an = a-(n - m)
(ii)
Where m and n are whole numbers and m We can generalize that if a is a non-zero integer or a non-zero rational number and m and n are positive integers, such that m n, then
am ÷ an = am - n if m m ÷ an = 1/an - m
Similarly, (a/b)m ÷ (a/b)n = (a/b)m -n
3. Power of a power
Example : (23)2, (52)6, (32 )-3
In power of a power you need multiply the powers.
In general, for any non-integer a, (am)n=am × n=amn.Thus where m and n are whole numbers.
If a is a non-zero rational number and m and n are positive integers, then {(a/b)m}n = (a/b) mn.
4. Multiplying power with the same exponents :
Example - 32 × 22, 53 × 73
We consider the product of 42 and 32, which have different bases, but the same exponents.
5. Negative Exponents :
If the exponent is negative we need to change it into positive exponent by writing the same in the denominator and 1 in the numerator.
If a is a non-zero integer or a non-zero rational number and m is a positive integers, thena-mis the reciprocal of am, i.e.,
a-m = 1/am, if we take a as p/q then (p/q)-m = 1/(p/q)m = (q/p)m
again, 1/a-m = am
Similarly, (a/b)-n = (b/a)n, where n is a positive integer.
6. Power with exponent Zero :
If the exponent is 0 then you get the result 1 whatever the base is.
example -80, ( a/b)0, m0.
If a is a non-zero integer or a non-zero rational number then,
a0 = 1
Similarly, (a/b)0 = 1
7. Fractional Exponents -
In fractional exponent we observe that the exponent is in fraction form.
Example - a1/n [Here a is called the base and 1/n is called the exponent or power]
= n√a [nth root of a]
=31/2= √3 [square root of 3]
=51/3 = ∛5 [cube root of 5
Multiplying powers with the same base
Example : x2 × x3, 23 × 25, (-3)2 × (-3)4
In multiplication of exponents if the bases are same then we need to add the exponents.
NOTE - (i) Exponents can be added only when the bases are same.
(ii) Exponents cannot be added if the bases are not same like.
2.Dividing powers with the same base
Example : 35 ÷ 31, 22 ÷ 21, 5(2) ÷ 53
In division if the bases are same then we need to subtract the exponents.
NOTE -
Where m and n are whole numbers and m n;
am ÷ an = am/an = a-(n - m)
(ii)
Where m and n are whole numbers and m We can generalize that if a is a non-zero integer or a non-zero rational number and m and n are positive integers, such that m n, then
am ÷ an = am - n if m m ÷ an = 1/an - m
Similarly, (a/b)m ÷ (a/b)n = (a/b)m -n
3. Power of a power
Example : (23)2, (52)6, (32 )-3
In power of a power you need multiply the powers.
In general, for any non-integer a, (am)n=am × n=amn.Thus where m and n are whole numbers.
If a is a non-zero rational number and m and n are positive integers, then {(a/b)m}n = (a/b) mn.
4. Multiplying power with the same exponents :
Example - 32 × 22, 53 × 73
We consider the product of 42 and 32, which have different bases, but the same exponents.
5. Negative Exponents :
If the exponent is negative we need to change it into positive exponent by writing the same in the denominator and 1 in the numerator.
If a is a non-zero integer or a non-zero rational number and m is a positive integers, thena-mis the reciprocal of am, i.e.,
a-m = 1/am, if we take a as p/q then (p/q)-m = 1/(p/q)m = (q/p)m
again, 1/a-m = am
Similarly, (a/b)-n = (b/a)n, where n is a positive integer.
6. Power with exponent Zero :
If the exponent is 0 then you get the result 1 whatever the base is.
example -80, ( a/b)0, m0.
If a is a non-zero integer or a non-zero rational number then,
a0 = 1
Similarly, (a/b)0 = 1
7. Fractional Exponents -
In fractional exponent we observe that the exponent is in fraction form.
Example - a1/n [Here a is called the base and 1/n is called the exponent or power]
= n√a [nth root of a]
=31/2= √3 [square root of 3]
=51/3 = ∛5 [cube root of 5
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