# 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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### 1 Answer

`sqrt(x*sqrt(x*sqrtx))`

To simplify, express the radicals as exponents.

`=sqrt(x*sqrt(x*x^(1/2)))`

`=sqrt(x*(x*x^(1/2))^(1/2))`

`=(x*(x*x^(1/2))^(1/2))^(1/2)`

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).

`=(x*(x^(1+1/2))^(1/2))^(1/2)`

`=(x(x^(3/2))^(1/2))^(1/2)`

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

`=(x*x^(3/2*1/2))^(1/2)`

`=(x*x^(3/4))^(1/2)`

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

`=(x^(1+3/4))^(1/2)`

`=(x^(7/4))^(1/2)`

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

`=x^(7/4*1/2)`

`=x^(7/8)`

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

`=root(8)((x^7))`

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