Question

I need to understand the steps and why. ty. rt{y} ^4" alt="\sqrt{y} ^4" align="absmiddle" class="latex-formula">

Answer 1

Answer:  $y^2$

Reason:

A square root represents an exponent of 1/2

So $\sqrt{y} = y^{1/2}$

Raising that to the fourth power gets us this,

$(\sqrt{y})^4 = (y^{1/2})^4\\\\ (\sqrt{y})^4 = y^{(1/2)*4}\\\\ (\sqrt{y})^4 = y^{2}$

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Edit: Here's an alternative route

$(\sqrt{y})^4 = (\sqrt{y})*(\sqrt{y})*(\sqrt{y})*(\sqrt{y})\\\\ (\sqrt{y})^4 = (\sqrt{y})^2*(\sqrt{y})^2\\\\ (\sqrt{y})^4 = y*y\\\\ (\sqrt{y})^4 = y^2$

The first line is possible because raising anything to the 4th power is the same as having 4 copies of it multiplied out (eg: 3^4 = 3*3*3*3 = 81).

Then in the second line, I collected the terms into pairs

By line 3, the square root and squaring exponent cancel each other out. This only works if y is not negative.