Question

if a + b + c is equals to zero then the value of root x power a into x power b into x power c ​

Answer 1

Basic rules for exponentiation

If n
n
is a positive integer and x
x
is any real number, then xn
x
n
corresponds to repeated multiplication
xn=x×x×⋯×xn times.
x
n
=
x
×
x
×
⋯
×
x
⏟
n
times
.
We can call this “x
x
raised to the power of n
n
,” “x
x
to the power of n
n
,” or simply “x
x
to the n
n
.” Here, x
x
is the base and n
n
is the exponent or the power.
From this definition, we can deduce some basic rules that exponentiation must follow as well as some hand special cases that follow from the rules. In the process, we'll define exponentials xa
x
a
for exponents a
a
that aren't positive integers.
The rules and special cases are summarized in the following table. Below, we give details for each one.
Rule or special case Formula Example
Product xaxb=xa+b
x
a
x
b
=
x
a
+
b
2223=25=32
2
2
2
3
=
2
5
=
32
Quotient xaxb=xa−b
x
a
x
b
=
x
a
−
b
2322=21=2
2
3
2
2
=
2
1
=
2
Power of power (xa)b=xab
(
x
a
)
b
=
x
a
b
(23)2=26=64
(
2
3
)
2
=
2
6
=
64
Power of a product (xy)a=xaya
(
x
y
)
a
=
x
a
y
a
36=62=(2⋅3)2=22⋅32=4⋅9=36
36
=
6
2
=
(
2
⋅
3
)
2
=
2
2
⋅
3
2
=
4
⋅
9
=
36
Power of one x1=x
x
1
=
x
21=2
2
1
=
2
Power of zero x0=1
x
0
=
1
20=1
2
0
=
1
Power of negative one x−1=1x
x
−
1
=
1
x
2−1=12
2
−
1
=
1
2
Change sign of exponents x−a=1xa
x
−
a
=
1
x
a
2−3=123=18
2
−
3
=
1
2
3
=
1
8
Fractional exponents xm/n=xm‾‾‾√n=(x√n)m
x
m
/
n
=
x
m
n
=
(
x
n
)
m
43/2=(4‾√)3=23=8
4
3
/
2
=
(
4
)
3
=
2
3
=
8
The rules
Product of exponentials with same base
If we take the product of two exponentials with the same base, we simply add the exponents:
xaxb=xa+b.(1)
(1)
x
a
x
b
=
x
a
+
b
.

To see this rule, we just expand out what the exponents mean. Let's start out with a couple simple examples.
3432=(3×3×3×3)×(3×3)=3×3×3×3×3×3=36
3
4
3
2
=(3×3×3×3)×(3×3) =3×3×3×3×3×3 =
3
6

y2y3=(y×y)×(y×y×y)=y×y×y×y×y=y5
y
2
y
3
=(y×y)×(y×y×y) =y×y×y×y×y =
y
5


The general case works the same way. We just need to keep track of the number of factors we have.
xaxb=x×⋯×xa times×x×⋯×xb times=x×⋯×xa+b times=xa+b
x
a
x
b
=
x
×
⋯
×
x
⏟
a
times
×
x
×
⋯
×
x
⏟
b
times
=
x
×
⋯
×
x
⏟
a
+
b
times
=
x
a
+
b


Quotient of exponentials with same base
If we take the quotient of two exponentials with the same base, we simply subtract the exponents:
xaxb=xa−b(2)
(2)
x
a
x
b
=
x
a
−
b


This rule results from canceling common factors in the numerator and denominator. For example:
y5y3=y×y×y×y×yy×y×y=(y×y)×(y×y×y)y×y×y=y×y=y2.
y
5
y
3
=
y
×
y
×
y
×
y
×
y
y
×
y
×
y
=
(
y
×
y
)
×
(
y
×
y
×
y
)
y
×
y
×
y
=y×y=
y
2
.

To show this in general, we look at two different cases. If we imagine that a>b
a
>
b
, then this rule follows from canceling the common b
b
factors of x
x
that occur in both the numerator and denominator. We are left with just b−a
b
−
a
factors of x
x
in the numerator.
xaxb=x×⋯×xa timesx×⋯×xb times=x×⋯×xa−b times×x×⋯×xb timesx×⋯×xb times=x×⋯×xa−b times=xa−b
x
a
x
b
=
x
×
⋯
×
x
⏞
a
times
x
×
⋯
×
x
⏟
b
times
=
x
×
⋯
×
x
⏞
a
−
b
times
×
x
×
⋯
×
x
⏞
b
times
x
×
⋯
×
x
⏟
b
times
=
x
×
⋯
×
x
⏟
a
−
b
times
=
x
a
−
b