Answer:
![x=19](https://tex.z-dn.net/?f=x%3D19)
Step-by-step explanation:
1) Use this rule:
.
![(x-3)^{\frac{1*1}{2*2} } =\sqrt{x-15}](https://tex.z-dn.net/?f=%28x-3%29%5E%7B%5Cfrac%7B1%2A1%7D%7B2%2A2%7D%20%7D%20%3D%5Csqrt%7Bx-15%7D)
2) Simplify 1 * 1 to 1.
![\sqrt[2*2]{x-3} =\sqrt{x-15}](https://tex.z-dn.net/?f=%5Csqrt%5B2%2A2%5D%7Bx-3%7D%20%3D%5Csqrt%7Bx-15%7D)
3) Simplify 2 * 2 to 4.
![\sqrt[4]{x-3} =\sqrt{x-15}](https://tex.z-dn.net/?f=%5Csqrt%5B4%5D%7Bx-3%7D%20%3D%5Csqrt%7Bx-15%7D)
4) Square both sides.
![\sqrt{x-3} =x-15](https://tex.z-dn.net/?f=%5Csqrt%7Bx-3%7D%20%3Dx-15)
5) Square both sides.
![x-3=x^2-30x+225](https://tex.z-dn.net/?f=x-3%3Dx%5E2-30x%2B225)
6) Move all terms to one side.
![x-3-x^2+30x-225=0](https://tex.z-dn.net/?f=x-3-x%5E2%2B30x-225%3D0)
7) Simplify
to
.
![31x-228-x^2=0](https://tex.z-dn.net/?f=31x-228-x%5E2%3D0)
8) Multiply both sides by -1.
![x^2-31x+228=0](https://tex.z-dn.net/?f=x%5E2-31x%2B228%3D0)
9) Factor
.
1) Ask: Which two numbers add up to -31 and multiply to 228?
and ![-12](https://tex.z-dn.net/?f=-12)
2) Rewrite the expression using the above.
![(x-19)(x-12)](https://tex.z-dn.net/?f=%28x-19%29%28x-12%29)
10) Solve for
.
1) Ask when will
equal zero?
When
or ![x-12=0](https://tex.z-dn.net/?f=x-12%3D0)
2) Solve each of the 2 equations above.
![x=19,12](https://tex.z-dn.net/?f=x%3D19%2C12)
11) Check solution.
When
2, the original equation
does not hold true. We will drop
from the solution set.
12) Therefore,
![x=19](https://tex.z-dn.net/?f=x%3D19)
Check the Answer:
![\sqrt{\sqrt{x-3}}=\sqrt{x-15}](https://tex.z-dn.net/?f=%5Csqrt%7B%5Csqrt%7Bx-3%7D%7D%3D%5Csqrt%7Bx-15%7D)
1) Let
.
![\sqrt{\sqrt{19-3}}=\sqrt{19-15}](https://tex.z-dn.net/?f=%5Csqrt%7B%5Csqrt%7B19-3%7D%7D%3D%5Csqrt%7B19-15%7D)
2) Simplify 19 - 3 to 16.
![\sqrt{\sqrt{16}}=\sqrt{19-15}](https://tex.z-dn.net/?f=%5Csqrt%7B%5Csqrt%7B16%7D%7D%3D%5Csqrt%7B19-15%7D)
3) Since 4 * 4 is 16 6, the square root of 16 is 4.
![\sqrt{4}=\sqrt{19-15}](https://tex.z-dn.net/?f=%5Csqrt%7B4%7D%3D%5Csqrt%7B19-15%7D)
4) Since 2 * 2 = 4, the square root of 4 is 2.
![2=\sqrt{19-15}](https://tex.z-dn.net/?f=2%3D%5Csqrt%7B19-15%7D)
5) Simplify 19 - 15 to 4.
![2=\sqrt{4}](https://tex.z-dn.net/?f=2%3D%5Csqrt%7B4%7D)
6) Since 2 * 2 = 4, the square root of 4 is 2.
![2 = 2](https://tex.z-dn.net/?f=2%20%3D%202)