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: $\sqrt[3]{a^2 \sqrt{a}}$.

Simplify fully the following expression with nested radicals, reducing it to a single radical and extracting all possible factors (assume that the variables are positive reals):

$\sqrt[4]{\frac{x^3 y}{z} \sqrt[3]{\frac{z^2}{x y^2} \sqrt{\frac{x}{y}}}}$.

Express the radicals $\sqrt[3]{5}$, $\sqrt[4]{10}$ and $\sqrt[6]{30}$ with a common index, and order them from smallest to largest, justifying your answer.

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

Multiply the following radicals with the same index and extract any possible factors from the final result: $\sqrt[4]{27a^2b^3} \times \sqrt[4]{3a^3b}$.

Calculate the power of the following radical and simplify the result by extracting all possible factors (assume positive variables): $(\sqrt[4]{8x^3y^2})^2$.

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

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: $\frac{2\sqrt{3} - \sqrt{2}}{\sqrt{3} + 2\sqrt{2}}$.

Expand the following expression using algebraic identities for the square of a difference and the product of binomials, and collect like terms: $(\sqrt{5} - 2\sqrt{3})^2 - (2\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})$.

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

Simplify the following algebraic expression completely, performing operations on the fractions using a common denominator and applying the properties of conjugate radicals:

$\frac{\sqrt{x+1} - \sqrt{x-1}}{\sqrt{x+1} + \sqrt{x-1}} + \frac{\sqrt{x+1} + \sqrt{x-1}}{\sqrt{x+1} - \sqrt{x-1}}$.

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.

The annual compound interest rate $r$ (expressed as a decimal) required for an initial principal $C_0$ to grow into a final amount $C_n$ after $n$ years is given by $r = \sqrt[n]{\frac{C_n}{C_0}} - 1$. Calculate the exact value of the rate $r$ if an investment multiplies its value by $\frac{5}{2}$ in exactly 3 years. Rationalise the resulting radical in your answer.

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.

Prove algebraically that the irrational number $x = \sqrt{7 + 4\sqrt{3}} + \sqrt{7 - 4\sqrt{3}}$ is actually an integer. Set up the equation $x^2$ and use algebraic identities to solve it.

In a regular pentagon of side length $L = 2$ cm, the length of any diagonal $D$ is related to the side length by the golden ratio $\Phi = \frac{1+\sqrt{5}}{2}$, satisfying $D = L \times \Phi$. Draw a horizontal diagonal to divide the pentagon into an upper isosceles triangle and a lower isosceles trapezium. Using Pythagoras' theorem, calculate the exact value of the total height $h$ of the pentagon by adding the heights of both shapes. Express the final result in the simplest possible form.

You can do it on paper and send a photo with your answer.

The Lorentz factor in special relativity theory is defined as $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$. If a subatomic particle travels at a speed of $v = \frac{\sqrt{5}}{3}c$, calculate the exact value of $\gamma$. Subsequently, determine the exact speed $v$ (in terms of $c$) at which it must travel for the factor to be $\gamma = \sqrt{2} + 1$, rationalising the result if necessary.

The maximum stress $S$ supported by the base of an architectural dome modelled using a conic section is given by the expression $S = \frac{P}{\sqrt{a^2 + b^2 - a\sqrt{b}}}$. If the structural load is $P = 14$ metric tons, with geometric parameters $a = \sqrt{2}$ and $b = 8$, calculate the exact value of the stress $S$, rationalising the final result.

Consider the Spiral of Theodorus, constructed using adjacent right-angled triangles. The first triangle has equal legs of length $a$. The nth triangle has legs given by the hypotenuse of the previous triangle and a new leg of length $a$. If the area of the sixth triangle in the sequence is exactly $A_6 = 12\sqrt{6}$ cm², calculate the exact value of the parameter $a$ and, with it, determine the exact perimeter of the third triangle in the spiral, simplifying the radicals.