Answer:
![{x}^{2} - 12x + 5 = 7](https://tex.z-dn.net/?f=%20%7Bx%7D%5E%7B2%7D%20%20-%2012x%20%2B%205%20%3D%207)
i) move constants to the right-hand side and change its sign
![{x}^{2} - 12 {x} = 7 - 5](https://tex.z-dn.net/?f=%20%7Bx%7D%5E%7B2%7D%20-%2012%20%7Bx%7D%20%3D%207%20-%205)
ii) subtract the numbers
![{x}^{2} - 12x = 2](https://tex.z-dn.net/?f=%20%7Bx%7D%5E%7B2%7D%20%20-%2012x%20%3D%202)
iii) add (12/2)² to both sides of the equation
![{x}^{2} - 12x + ( \frac{12}{2} ) {}^{2} = 2 + ( \frac{12}{2} ) {}^{2}](https://tex.z-dn.net/?f=%20%7Bx%7D%5E%7B2%7D%20%20-%2012x%20%2B%20%28%20%5Cfrac%7B12%7D%7B2%7D%20%29%20%7B%7D%5E%7B2%7D%20%20%3D%202%20%2B%20%28%20%5Cfrac%7B12%7D%7B2%7D%20%29%20%7B%7D%5E%7B2%7D%20)
iv) using a²-2ab+b²=(a-b)² , factorize the expression
![(x - \frac{12}{2} ) {}^{2} = 2 + ( \frac{12}{2} ) {}^{2}](https://tex.z-dn.net/?f=%28x%20-%20%20%5Cfrac%7B12%7D%7B2%7D%20%29%20%7B%7D%5E%7B2%7D%20%20%3D%202%20%2B%20%28%20%5Cfrac%7B12%7D%7B2%7D%20%29%20%7B%7D%5E%7B2%7D%20)
v) calculate the value
![(x - \frac{12}{2}) {}^{2} = 2 + 36](https://tex.z-dn.net/?f=%28x%20-%20%20%5Cfrac%7B12%7D%7B2%7D%29%20%7B%7D%5E%7B2%7D%20%20%20%3D%202%20%2B%2036)
![(x - \frac{12}{2}) {}^{2} = 38](https://tex.z-dn.net/?f=%28x%20-%20%20%5Cfrac%7B12%7D%7B2%7D%29%20%7B%7D%5E%7B2%7D%20%3D%2038)
vi) reduce the fraction
![(x - 6) {}^{2} = 38](https://tex.z-dn.net/?f=%28x%20-%206%29%20%7B%7D%5E%7B2%7D%20%20%3D%2038)
vii) solve the equation for x
![x - 6 = + - \sqrt{38}](https://tex.z-dn.net/?f=x%20-%206%20%3D%20%20%2B%20%20-%20%20%5Csqrt%7B38%7D%20)
1) first value of x
![x - 6 = \sqrt{38}](https://tex.z-dn.net/?f=x%20-%206%20%3D%20%20%5Csqrt%7B38%7D%20)
![x = \sqrt{38} + 6 \: or \: 12.16](https://tex.z-dn.net/?f=x%20%3D%20%20%5Csqrt%7B38%7D%20%20%2B%206%20%5C%3A%20or%20%5C%3A%2012.16)
2) second value of x
![x - 6 = - \sqrt{38}](https://tex.z-dn.net/?f=x%20-%206%20%3D%20%20-%20%20%5Csqrt%7B38%7D%20)
![x = - \sqrt{38} + 6 \: or \: - 0.16](https://tex.z-dn.net/?f=x%20%3D%20%20-%20%20%5Csqrt%7B38%7D%20%20%20%2B%206%20%5C%3A%20or%20%5C%3A%20%20-%200.16)
Given
The equation is in the form
![=\frac{4^{\sqrt{400}}}{4^{\sqrt{5}}}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B4%5E%7B%5Csqrt%7B400%7D%7D%7D%7B4%5E%7B%5Csqrt%7B5%7D%7D%7D)
Simplify the above equation
To proof
As given in the question the equation is written in the form
![=\frac{4^{\sqrt{400}}}{4^{\sqrt{5}}}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B4%5E%7B%5Csqrt%7B400%7D%7D%7D%7B4%5E%7B%5Csqrt%7B5%7D%7D%7D)
By using the exponent property
when you divide powers with the same base you just have to subtract the exponents. i.e
![\frac{y^{a}}{y^{b}} = y^{a-b}](https://tex.z-dn.net/?f=%5Cfrac%7By%5E%7Ba%7D%7D%7By%5E%7Bb%7D%7D%20%3D%20y%5E%7Ba-b%7D)
y is not equal to zero.
Now apply this property to the above equation
![=4^{\sqrt{400}- \sqrt{5}}](https://tex.z-dn.net/?f=%3D4%5E%7B%5Csqrt%7B400%7D-%20%5Csqrt%7B5%7D%7D)
As
![\sqrt{400} = 20](https://tex.z-dn.net/?f=%5Csqrt%7B400%7D%20%3D%2020)
now put this in the above equation
we get
![=4^{20- \sqrt{5}}](https://tex.z-dn.net/?f=%3D4%5E%7B20-%20%5Csqrt%7B5%7D%7D)
Hence proved
Hi again
For m = -6
Evaluate:
m² + 5m - 1
(-6)² + 5(-6) - 1
36 - 30 - 1
= 6 - 1
= 5
Let me know if you have any questions. As always, it is my pleasure to help students like you!