Answer:
[tex]x=19[/tex]
Step-by-step explanation:
1) Use this rule: [tex](x^a)^b=x^a^b[/tex] .
[tex](x-3)^{\frac{1*1}{2*2} } =\sqrt{x-15}[/tex]
2) Simplify 1 * 1 to 1.
[tex]\sqrt[2*2]{x-3} =\sqrt{x-15}[/tex]
3) Simplify 2 * 2 to 4.
[tex]\sqrt[4]{x-3} =\sqrt{x-15}[/tex]
4) Square both sides.
[tex]\sqrt{x-3} =x-15[/tex]
5) Square both sides.
[tex]x-3=x^2-30x+225[/tex]
6) Move all terms to one side.
[tex]x-3-x^2+30x-225=0[/tex]
7) Simplify [tex]x-3-x^2+30x-225[/tex] to [tex]31x-228-x^2[/tex].
[tex]31x-228-x^2=0[/tex]
8) Multiply both sides by -1.
[tex]x^2-31x+228=0[/tex]
9) Factor [tex]x^2-31x+228[/tex].
1) Ask: Which two numbers add up to -31 and multiply to 228?
[tex]-19[/tex] and [tex]-12[/tex]
2) Rewrite the expression using the above.
[tex](x-19)(x-12)[/tex]
10) Solve for [tex]x[/tex].
1) Ask when will [tex](x-19)(x-12)[/tex] equal zero?
When [tex]x-19 =0[/tex] or [tex]x-12=0[/tex]
2) Solve each of the 2 equations above.
[tex]x=19,12[/tex]
11) Check solution.
When [tex]x=12[/tex] 2, the original equation [tex]\sqrt{\sqrt{x-3}}=\sqrt{x-15}[/tex] does not hold true. We will drop [tex]x=12[/tex] from the solution set.
12) Therefore,
[tex]x=19[/tex]
Check the Answer:
[tex]\sqrt{\sqrt{x-3}}=\sqrt{x-15}[/tex]
1) Let [tex]x=19[/tex].
[tex]\sqrt{\sqrt{19-3}}=\sqrt{19-15}[/tex]
2) Simplify 19 - 3 to 16.
[tex]\sqrt{\sqrt{16}}=\sqrt{19-15}[/tex]
3) Since 4 * 4 is 16 6, the square root of 16 is 4.
[tex]\sqrt{4}=\sqrt{19-15}[/tex]
4) Since 2 * 2 = 4, the square root of 4 is 2.
[tex]2=\sqrt{19-15}[/tex]
5) Simplify 19 - 15 to 4.
[tex]2=\sqrt{4}[/tex]
6) Since 2 * 2 = 4, the square root of 4 is 2.
[tex]2 = 2[/tex]
