Answer:
-1
Step-by-step explanation:
We need to simplify the given expression . The given expression is ,
[tex]\rm\implies ( 2 \sqrt6 +5)^{2n-1} ( 2\sqrt6 - 5)^{2n-1}[/tex]
Here we can see that the power of the both exponent is same that is (2n+1) . Recall the property of exponents ,
[tex]\bf\implies a^m \times b^m = (ab)^m [/tex]
Using this property , we have ,
[tex]\rm\implies ( 2 \sqrt6 +5)^{2n-1} ( 2\sqrt6 - 5)^{2n-1}[/tex]
This can be written as ,
[tex]\rm\implies\{( 2 \sqrt6 +5) ( 2\sqrt6 - 5)\}^{2n-1}[/tex]
Simplifying using ( a+b)(a-b) = a² - b² ,
[tex]\rm\implies\{( 2 \sqrt6)^2 -5^2 \}^{2n-1}\\\\\rm\implies ( 24 - 25)^{2n-1} [/tex]
Subtracting the numbers inside the brackets ,
[tex]\rm\implies (-1)^{2n - 1 }[/tex]
Now we know that every odd number is in the form of 2n -1 , where n is any integer. Therefore , the power is odd .
Since the base is (-1) , for even power it is 1 and for odd power it is -1 . Therefore the final answer is ,
[tex]\rm\implies\boxed{\quad \red{ -1 }\quad }[/tex]
Hence the required answer is (-1) .
