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:
Ax * Ay = Ax+y
(Ax)y = Ax*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
$${\sqrt{{\mathtt{a}}}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{b}}}} = {\sqrt{{\mathtt{ab}}}}$$
$${\sqrt{{{\mathtt{x}}}^{{\mathtt{10}}}}}{\mathtt{\,\times\,}}{\sqrt{{{\mathtt{x}}}^{{\mathtt{4}}}}} = {\sqrt{{{\mathtt{x}}}^{{\mathtt{10}}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{4}}}}} = {\sqrt{{{\mathtt{x}}}^{{\mathtt{14}}}}} = {\sqrt{{\left({{\mathtt{x}}}^{{\mathtt{7}}}\right)}^{{\mathtt{2}}}}} = {{\mathtt{x}}}^{{\mathtt{7}}}$$
.In the form of a formula this is:
$${\sqrt{{{\mathtt{x}}}^{{\mathtt{10}}}}}{\mathtt{\,\times\,}}{\sqrt{{{\mathtt{x}}}^{{\mathtt{4}}}}}$$
Some things to consider:
Ax * Ay = Ax+y
(Ax)y = Ax*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