Question

Simplify this equation
Thanks a lot!!!

Answer 1

-1

Explanation:

We need to simplify the given expression . The given expression is ,

`\rm\implies ( 2 \sqrt6 +5)^(2n-1) ( 2\sqrt6 - 5)^(2n-1)`

Here we can see that the power of the both exponent is same that is (2n+1) . Recall the property of exponents ,

`\bf\implies a^m * b^m = (ab)^m`

Using this property , we have ,

`\rm\implies ( 2 \sqrt6 +5)^(2n-1) ( 2\sqrt6 - 5)^(2n-1)`

This can be written as ,

`\rm\implies\{( 2 \sqrt6 +5) ( 2\sqrt6 - 5)\}^(2n-1)`

Simplifying using ( a+b)(a-b) = a² - b² ,

`\rm\implies\{( 2 \sqrt6)^2 -5^2 \}^(2n-1)\\\\\rm\implies ( 24 - 25)^(2n-1)`

Subtracting the numbers inside the brackets ,

`\rm\implies (-1)^(2n - 1 )`

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 ,

`\rm\implies\boxed{\quad \red{ -1 }\quad }`

Hence the required answer is (-1) .