Title:-What are Powers in Math
Introduction:-
What are Powers in Math or we can say what are exponents in math is our today's main discussion which is a very nice and an important topic of mathematics.
Explanation:-
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To start this discussion let us quick remember in general we find four operations in algebra and they are addition,subtraction,multiplication and division. Powers are associated with multiplication.Let us know, What are Powers in Math.
Let,
us suppose, aⁿ is an expression ,here 'a' is called base and n is called exponent and aⁿ is the power of a in the expression.In another way we can explain that the total number of times when any number is multiplied itself is called its power .
e.g;
3 × 3 × 3 × 3 × 3 = 3⁵
where, 5 is the exponent of base 3 and 3⁵ is the power of 3
Now we will know about formula of power.
The formulae of exponents :-
(i) aⁿ × aⁱ = aⁿ⁺ⁱ
Rule(1):-
In case of multiplication of exponents, If the bases are equal then powers are added.
(ii) aⁿ ÷ aⁱ = aⁿ⁻ⁱ
Rule(2):-
In case of division of exponents,If the bases are equal then powers are subtracted.
(iii) a⁰ = 1
Rule(3):-
If any number has power zero, then result will be 1
(iv) a⁻ⁿ = 1/aⁿ
Rule(4):-
If any number has power negative,then it will be 1 divides the number withe power to make it positive.
(v) (aⁿ)ⁱ = aⁿⁱ
Rule(5):-
In case of exponents if any expression has the power of power, then simply powers are multiplied themselves.
(vi)√a = a¹⁽²
Rule(6):-
A surd or root number can be converted into exponent by dividing the power with root value.
Let, us now we will prove some above formulas of exponents.
Prove that,aⁿ × aⁱ = aⁿ⁺ⁱ
Answer:-
L.H.S = aⁿ × aⁱ
=(a×a×a×a…..n number of factors)×(a×a×a×a....i number of factors)
= a × a × a × a ........(n+i) number of fectors
= aⁿ⁺ⁱ
= R.H.S Proved.
Prove that,aⁿ ÷ aⁱ = aⁿ⁻ⁱ
Answer:-
L.H.S = aⁿ ÷ aⁱ
(a×a×a…..n number of factors)
= —-----------------------------------------
(a×a×a…..i number of factors)
= a×a×a..... (n-i) number of fectors
= aⁿ⁻ⁱ
= R.H.S Proved.
Prove that, a⁰ = 1
Answer:-
L.H.S = a⁰
= aⁿ⁻ⁿ
aⁿ
= —-------
aⁿ
[ ∵ Exponents are in subtraction, so they are will be in division individually.]
= 1
= R.H.S Proved.
Prove that, a⁻ⁿ = 1/aⁿ
Answer:-
L.H.S = a⁻ⁿ
= a⁰⁻ⁿ
a⁰
= —--
aⁿ
[ ∵ Exponents are in subtraction, so they are will be in division individually.]
1
= —----- [∵using formula(iii)
aⁿ
= R.H.S Proved.
Some problems related on exponents:
Express the following numbers in exponents or indices form.
(i) 64 (ii) 343
Answer:- (i) 64
= 4³ [∵ 4 × 4 × 4 = 64 ]
= (2²)³ [ ∵ 4 = 2 × 2 ]
= 2⁶ [ ∵ According to the law of exponents power of power is multiplied themselves]
(ii) 343
= 7³ [∵ 7 × 7 × 7 = 343]
Simplify:-
(i) 125⁻¹⁽³ (ii) 64¹⁽⁶
Answer:-
(i) 125⁻¹⁽³
= {(5)³}⁻¹⁽³ [ ∵5×5×5 = 125]
= 5 ³×⁻¹⁽³ [ ∵power of power will be multiplied]
= 5⁻¹ [ ∵3×(- ⅓ )= -1]
= ⅕ [ using formula (iv)
(ii) 64¹⁽⁶
= (2⁶)¹⁽⁶
= 2⁶ ×¹⁽⁶ [ ∵power of power will be multiplied]
= 2¹ [∵ 6 ×1/6 =1]
= 2
Find the value of n from the given expression:-
Answer:- 27ⁿ ÷ ⅓ = 3⁵
=> (3³)ⁿ ÷ 3⁻¹ = 3⁵
=> 3³ⁿ ÷ 3⁻¹ =3⁵[∵using formula(v)]
=> 3³ⁿ ⁻ ¹ = 3⁵ [∵using formula(ii)]
=> 3n - 1 = 5 [ ∵ bases in both sides are equal, so their powers are also equal]
=>3n = 5 + 1
=> n = 6/3
=> n =2
Conclusion:-
From the above discussion we have learnt
What are Powers in Math. Generally exponents has six basic formulas using.......
FREQUENTLY ASKED QUESTIONS:-
Difference between exponent and power with example
Answer:- The difference between exponent and power :-
The total number of times when any number is multiplied itself is called its power . e.g; 2 ×2×2 =2³, where 2³ is the power of 2
On the othe hand the number which is raised to the base factor/ number is called exponent. e.g; In 2 ×2×2 =2³ , base is 2 and exponent is 3.
Exponents and Powers Class 10
Answer:- The number which is raised to the base factor/ number is called exponent. e.g; In 2 ×2×2 =2³ , base is 2 and exponent is 3.
The total number of times when any number is multiplied itself is called its power . e.g; 2 ×2×2 =2³, where 2³ is the power of 2
Exponent symbol name
Answer:- The name of exponent symbol is caret(^)
What does power mean in math?
Answer:- Power mean in math that the total number of times when any number is multiplied itself is called its power . e.g; 2 ×2×2 =2³, where 2³ is the power of 2
What is 625 as a power of 5?
Answer:- 625 = 5 × 5 × 5 × 5 = 5⁴
Difference between exponent and power with example
Answer:- The difference between exponent and power :- The total number of times when any number is multiplied itself is called its power . e.g; 2 ×2×2 =2³, where 2³ is the power of 2 On the othe hand the number which is raised to the base factor/ number is called exponent. e.g; In 2 ×2×2 =2³ , base is 2 and exponent is 3.
Exponents and Powers Class 10
Answer:- The number which is raised to the base factor/ number is called exponent. e.g; In 2 ×2×2 =2³ , base is 2 and exponent is 3. The total number of times when any number is multiplied itself is called its power . e.g; 2 ×2×2 =2³, where 2³ is the power of 2
Exponent symbol name
Answer:- The name of exponent symbol is caret(^)
What does power mean in math?
Answer:- Power mean in math that the total number of times when any number is multiplied itself is called its power . e.g; 2 ×2×2 =2³, where 2³ is the power of 2
What is 625 as a power of 5?
Answer:- 625 = 5 × 5 × 5 × 5 = 5⁴
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