Equations with radicals

Solve the following radical equation and check the validity of the solution obtained:

$\sqrt{3x - 2} = 4$

Determine the value of $x$ satisfying the following irrational equation:

$\sqrt{2x + 7} - x = 2$

Solve the following equation containing a higher-order root:

$\sqrt[3]{5x - 4} = -1$

Find the values of $x$ satisfying the following equation with fractions inside the radicand:

$\sqrt{\frac{4x - 1}{3}} = 5$

Find the solution to the equation by equating both radicals and check the result:

$\sqrt{x + 5} = \sqrt{3x - 1}$

Solve the following irrational equation containing two radicals:

$\sqrt{x + 4} - \sqrt{x - 1} = 1$

Find the real solutions to the following equation containing a nested radical (a radical inside another):

$\sqrt{x + \sqrt{x + 3}} = 3$

Solve the following irrational equation with roots of different indices. It will be very helpful to perform the substitution $t = \sqrt[6]{x}$:

$\sqrt{x} - 7\sqrt[3]{x} + 12\sqrt[6]{x} = 0$

Solve the following irrational equation containing three radicals. Take special care when collecting terms after squaring for the first time:

$\sqrt{x + 2} + \sqrt{x - 3} = \sqrt{3x + 4}$

Solve the following equation, which combines the properties of absolute values with irrational expressions. Carefully consider the cases for the absolute value:

$|\sqrt{3x + 1} - 4| = 2$

The base radius $r$ of a cylinder of volume $V$ and height $h$ is calculated using the expression $r = \sqrt{\frac{V}{\pi h}}$. If a cylindrical container of height $15\text{ cm}$ has a volume of $471\text{ cm}^3$, set up the corresponding equation and find its base radius (use the approximation $\pi \approx 3,14$).

Calculate the $x$-coordinate where a point $P(x, 4)$ must be placed relative to point $A(-2, 1)$ so that the Euclidean distance between them is exactly $5$ units. Set up the corresponding radical equation using the distance formula and solve it.

Special relativity establishes that the relativistic mass $m$ of a moving object is given by the equation $m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $m_0$ is the rest mass, $v$ is the velocity of the object and $c$ is the speed of light in a vacuum. Rearrange algebraically to find the velocity $v$ in terms of $m$, $m_0$ and $c$. Subsequently, calculate at what fraction of the speed of light $c$ a spacecraft must travel for its relativistic mass to be exactly double its rest mass.