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It does not come out to y^7 FYI. I can't figure it out

Guest Jul 12, 2017
#1
+7155
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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...

$$=7\,\cdot\,\sqrt{3}\,\cdot\,x \,\cdot\, x \,\cdot\,x \,\cdot \, y\,\cdot\,y\,\cdot\,y\,\cdot\,\sqrt{y} \\~\\ =7\,\cdot\,\sqrt3\,\cdot\,x^3\,\cdot\,y^3\,\cdot\,\sqrt{y} \\~\\ =7x^3y^3\sqrt{3y}$$

hectictar  Jul 12, 2017