simplify the expression "the square root of x^10 times the square root of x^4" completely

In the form of a formula this is:

  $${\sqrt{{{\mathtt{x}}}^{{\mathtt{10}}}}}{\mathtt{\,\times\,}}{\sqrt{{{\mathtt{x}}}^{{\mathtt{4}}}}}$$

Some things to consider:

A^x * A^y = A^x+y

(A^x)^y = A^x*y

And Square-root of x is equal to x^(1/2)

So this forumula is equivalent to:

(x^10)^¹/₂ * (x^4)^¹/₂

= x^(10/2) * x^(4/2)

= x^5 * x^2

= x^(5 + 2)

= X^7

$${\sqrt{{{\mathtt{x}}}^{{\mathtt{10}}}}}{\mathtt{\,\times\,}}{\sqrt{{{\mathtt{x}}}^{{\mathtt{4}}}}} = {\sqrt{{\left({{\mathtt{x}}}^{{\mathtt{5}}}\right)}^{{\mathtt{2}}}}}{\mathtt{\,\times\,}}{\sqrt{{\left({{\mathtt{x}}}^{{\mathtt{2}}}\right)}^{{\mathtt{2}}}}} = {{\mathtt{x}}}^{{\mathtt{5}}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}} = {\mathtt{x}}{\mathtt{\,\times\,}}\left({\mathtt{5}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}\right) = {{\mathtt{x}}}^{{\mathtt{7}}}$$

Second way

$${\sqrt{{\mathtt{a}}}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{b}}}} = {\sqrt{{\mathtt{ab}}}}$$

$${\sqrt{{{\mathtt{x}}}^{{\mathtt{10}}}}}{\mathtt{\,\times\,}}{\sqrt{{{\mathtt{x}}}^{{\mathtt{4}}}}} = {\sqrt{{{\mathtt{x}}}^{{\mathtt{10}}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{4}}}}} = {\sqrt{{{\mathtt{x}}}^{{\mathtt{14}}}}} = {\sqrt{{\left({{\mathtt{x}}}^{{\mathtt{7}}}\right)}^{{\mathtt{2}}}}} = {{\mathtt{x}}}^{{\mathtt{7}}}$$