Question

if x=10^a , y=10^b and x^b.y^a=100 , then prove that xy=1​

Answer 1

I have attached the answer here

Answer 2

Step-by-step explanation:

Correction :-

Prove that ab = 1

Given :-

X= 10^a ,

Y = 10^b,

X^b . Y^a = 100

To find :-

Prove that : ab= 1

Solution :-

Method-1:-

Given that :

X= 10^a -------(1)

Y = 10^b --------(2)

X^b . Y^a = 100----(3)

On Substituting the values of X and Y in (3) then

=> (10^a)^b . (10^b)^a = 100

=> 10^(ab) . 10^(ba) = 100

Since (a^m)^n = a^(mn)

=> 10^(ab+ab) = 100

Since a^m×a^n = a^(m+n)

=> 10^(2ab) = 100

=> 10^(2ab) = 100

=> (10²)^(ab) = 100

=> (100)^(ab) = 100¹

=> ab = 1

Hence, Proved.

Used formulae:-

• (a^m)^n = a^(mn)

• a^m×a^n = a^(m+n)

Or if the question is like

If a = 10^x and b = 10^y and a^y .b^x = 100 then prive that xy = 1

Then

a = 10^x

b = 10^y

a^y.b^x = 100

=> (10^x)^y . (10^y)^x = 100

=> 10^(xy). 10^(yx) = 100

=> 10^(xy+xy) = 100

=> 10^(2xy) = 100

=> (10^2)^(xy) = 100

=> (100)^xy = 100¹

=> xy = 1

Hence, Proved.