If `x gt= 0` , then `sqrt(x* sqrt(x* sqrt(x)))`  equals:a. `root(8)((x^7))` b. `xroot(4)(x)` c. `x sqrt(x)` d. `sqrt(x^3)`

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lemjay | High School Teacher | (Level 3) Senior Educator

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To simplify, express the radicals as exponents.




Since the resulting expression is nested parenthesis, start to simplify the innermost expression.

To multiply x*x^1/2, apply the properties of exponent which is a^m*a^n=a^(m+n).



Then, simplify `(x^(3/2))^(1/2)` . To do so, apply  `(a^m)^n=a^(m*n)` .



Next simplify x*x^(3/4). Again, apply the rule of multiplying same base.



And, apply `(a^m)^n = a^(m*n)` , too.



Since the exponent is a fraction, express it as a radical. Note that `a^(m/n)= root(n)(a^m)` .


Hence,  `sqrt(x*sqrt(x*sqrtx))=root(8)((x^7))` .

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