Property 7
\(\sqrt[n]{x^n}=\begin{cases}x&\text{(if n is odd)}\\|x|&\text{(if n is even)}\end{cases}\)
If the power of the radical is even than the result is always positive.
1.
\(x<0\)
\(\Rightarrow\sqrt{x^2}=?\)
2.
\(x<0<y\)
\(\Rightarrow\sqrt{x^2}+\sqrt[4]{y^4}=?\)
3.
\(x>0\)
\(\Rightarrow\sqrt[3]{(-x)^3}=?\)
4.
\(x<0<y\)
\(\Rightarrow\sqrt{(x-y)^2}=?\)
5.
\(x<0<y\)
\(\Rightarrow\sqrt{x^2}+\sqrt{y^2}=?\)
6.
\(x<0<y\)
\(\Rightarrow\sqrt[3]{(-x)^3}+\sqrt{(x-y)^2}=?\)
7.
\(0<x<3\)
\(\Rightarrow\sqrt{x^2-8x+16}+\sqrt[4]{x^4}=?\)
8.
\(1<x<3\)
\(\Rightarrow\sqrt{x^2-2x+1}+\sqrt{x^2-6x+9}=?\)
9.
\(b<0<a\)
\(\Rightarrow\sqrt{a^2-2ab+b^2}+\sqrt{b^2}=?\)
10.
\(a=\sqrt3+1\)
\(b=\sqrt3-2\)
\(\Rightarrow\sqrt{(b-a)^2}=?\)
11.
\(3<x<4\)
\(\Rightarrow\sqrt{x^2-5x+5+\sqrt{x^2-8x+16}}=?\)
12.
\(1<x<2\)
\(\Rightarrow\sqrt{x^2-x-1+\sqrt{x^2-4x+4}}=?\)
13.
\(1<x<3\)
\(\Rightarrow\sqrt{x^2-x-2+\sqrt{x^2-6x+9}}=?\)