Sometimes when these exponents get confusing it helps to write out all the letters as many times as the exponent says. Also, we can write out the prime factorization 147...
\(\sqrt{{\color{magenta}147}\,\cdot\,{\color{RedOrange}x^6}\,\cdot\,{\color{teal}y^7}} \\~\\ =\sqrt{{\color{magenta}7\,\cdot\,7\,\cdot\,3}\,\cdot\,{\color{RedOrange}x\,\cdot\,x\,\cdot\,x\,\cdot\,x\,\cdot\,x\,\cdot\,x}\,\cdot\,{\color{teal}y\,\cdot\,y\,\cdot\,y\,\cdot\,y\,\cdot\,y\,\cdot\,y\,\cdot\,y}} \\~\\ =\sqrt{{\color{magenta}7\,\cdot\,7}}\cdot\sqrt{{\color{magenta}3}}\,\cdot\,\sqrt{{\color{RedOrange}x \,\cdot\, x}} \,\cdot\, \sqrt{{\color{RedOrange}x \,\cdot\, x}}\,\cdot\, \sqrt{{\color{RedOrange}x \,\cdot\, {\color{RedOrange}x}}}\,\cdot\,\sqrt{{\color{teal}y\,\cdot\,y}}\,\cdot\,\sqrt{{\color{teal}y\,\cdot\,y}}\,\cdot\,\sqrt{{\color{teal}y\,\cdot\,y}}\,\cdot\,\sqrt{{\color{teal}y}}\)
The square root of (7 * 7) is the square root of 7 squared, which is 7 .
The square root of (x * x) is x . The square root of (y * y) is y .
So we can write the original expression as...
\( =7\,\cdot\,\sqrt{3}\,\cdot\,x \,\cdot\, x \,\cdot\,x \,\cdot \, y\,\cdot\,y\,\cdot\,y\,\cdot\,\sqrt{y} \\~\\ =7\,\cdot\,\sqrt3\,\cdot\,x^3\,\cdot\,y^3\,\cdot\,\sqrt{y} \\~\\ =7x^3y^3\sqrt{3y}\)
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