Radicals

Express as a power with a fractional index and simplify fully the expression $\sqrt[6]{x^4 y^2 z^8}$, assuming that all variables represent positive real numbers.

Determine whether the radicals $\sqrt[4]{25x^2}$ and $\sqrt{5x}$ are equivalent for any value of $x > 0$. Justify your method using fractional indices.

Take out as many factors as possible from inside the following radical, simplifying the expression fully: $\sqrt[3]{108 a^5 b^7 c^3}$.

Simplify the following expression by applying the root of a root property and expressing the result with a single radical: $\sqrt[3]{a^2 \sqrt{a}}$.

Perform the additions and subtractions of the following like radicals, first extracting factors to simplify the result: $5\sqrt{18} - 2\sqrt{50} + \sqrt{8}$.

Carry out the following mixed operation with radicals of different indices, expressing the final result as a single simplified radical: $\frac{\sqrt{2x} \cdot \sqrt[3]{4x^2}}{\sqrt[6]{8x^5}}$.

Rationalise the denominator of the following expression and simplify the final result, extracting factors if possible: $\frac{6x}{\sqrt[5]{8x^2y^3}}$.

Rationalise the denominator by multiplying by the conjugate expression and simplify by expanding the numerator: $\frac{2\sqrt{3} - \sqrt{2}}{\sqrt{3} + 2\sqrt{2}}$.

Rationalise the denominator of the following fraction containing three radical terms and simplify the final result: $\frac{4}{\sqrt{2} + \sqrt{3} - \sqrt{5}}$.

The period $T$ of a simple pendulum is given by the physics formula $T = 2\pi\sqrt{\frac{L}{g}}$, where $L$ is the length of the string and $g$ is acceleration due to gravity. If an engineer needs the new period to be exactly $\sqrt{2} + 1$ times the original period, by what exact factor must they multiply the original length $L$? Express your reasoning algebraically by expanding the square of the binomial.

In a rectangular prism, the lengths of its three concurrent edges measure exactly $\sqrt{2}$ m, $\sqrt{3}$ m and $\sqrt{5}$ m. Calculate the exact value of the space diagonal of the prism ($D = \sqrt{a^2 + b^2 + c^2}$) and its volume, expressing both results in their simplest radical form.

According to Kleiber's Law in biology, the basal metabolic rate $R$ of a mammal scales with its body mass $M$ according to the relationship $R = k \sqrt[4]{M^3}$, where $k$ is a constant. If animal $A$ has a mass exactly $27$ times greater than that of animal $B$, determine the exact ratio $\frac{R_A}{R_B}$ using properties of radicals and extracting factors.