Question

given all variables below are positive, express in simplest radical form with a rational denominator.\\(dfrac{sqrt{1008x^{6}}}{sqrt{9x^{3}}}\\)

Explanation:

Step1: Combine the radicals

Using the property $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$, we can rewrite the expression as $\sqrt{\frac{1008x^6}{9x^3}}$.

Step2: Simplify the fraction inside the radical

Simplify the coefficients and the variables separately. For the coefficients, $\frac{1008}{9} = 112$. For the variables, using the quotient rule of exponents $x^m\div x^n = x^{m - n}$, we have $x^6\div x^3 = x^{6 - 3}=x^3$. So now we have $\sqrt{112x^3}$.

Step3: Factor the radicand

Factor 112 as $16\times7$ and $x^3$ as $x^2\times x$. So $\sqrt{112x^3}=\sqrt{16\times7\times x^2\times x}$.

Step4: Simplify the square root

Using the property $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ (where $a\geq0$ and $b\geq0$), we get $\sqrt{16}\times\sqrt{x^2}\times\sqrt{7x}$. Since $\sqrt{16} = 4$ and $\sqrt{x^2}=x$ (because $x$ is positive), this simplifies to $4x\sqrt{7x}$.

Answer:

$4x\sqrt{7x}$