Powers with rational exponents. Radicals

Express the following powers with fractional exponents as radicals and calculate their exact value if it is an integer:

• a) $27^{\frac{1}{3}}$

• b) $16^{\frac{3}{4}}$

• c) $5^{\frac{2}{3}}$

Write the following radicals in the form of a power with a fractional exponent and simplify the fraction in the exponent:

• a) $\sqrt[6]{x^2}$

• b) $\sqrt[15]{a^5}$

• c) $\sqrt[8]{2^6}$

Take out all possible factors from inside the following radicals:

• a) $\sqrt{50}$

• b) $\sqrt[3]{16}$

• c) $\sqrt[4]{x^7 y^5}$

Insert the factors into the radical and simplify the resulting expression if possible:

• a) $3 \sqrt{2}$

• b) $x^2 \sqrt[3]{x}$

• c) $2a \sqrt[4]{3a^2}$, with $a \ge 0$

Calculate the following products of radicals of the same index and take out factors from the result if possible:

• a) $\sqrt{3} \cdot \sqrt{12}$

• b) $\sqrt[3]{4} \cdot \sqrt[3]{10}$

Calculate and simplify the result of the following divisions of radicals:

• a) $\frac{\sqrt{72}}{\sqrt{2}}$

• b) $\frac{\sqrt[3]{40}}{\sqrt[3]{5}}$

Calculate the following powers of radicals, moving the exponent inside the radicand and expressing the result in the simplest form possible:

• a) $(\sqrt[3]{5})^6$

• b) $(\sqrt{x^3})^4$

Express as a single radical using the root of a root rule and simplify the index if possible:

• a) $\sqrt{\sqrt[3]{64}}$

• b) $\sqrt[4]{\sqrt{x^8}}$

For the manufacture of three electronic components, protective sheets are required whose thicknesses, measured in millimetres, are given by the expressions $A = \sqrt[3]{5}$, $B = \sqrt[4]{7}$ and $C = \sqrt[6]{24}$. Express the radicals with a common index to order the sheet thicknesses from smallest to largest.

Simplify the following expression with nested radicals by progressively moving factors into the innermost radicals and expressing the final result as a single root:

In a biology laboratory, the number of bacteria in a given culture follows the expression $N(t) = 5000 \cdot 8^{\frac{t}{3}}$, where $t$ is the time elapsed in hours. Express this formula using radicals and calculate the exact number of bacteria present after $2$ hours.

A circular town square has an area of $120\pi$ m$^2$. Given that the area of a circle is given by the formula $A = \pi r^2$, calculate the exact radius of the square, fully simplifying the radical.

Kepler's third law relates the orbital period $T$ of a planet to the mean radius of its orbit $r$ using the expression $T = k^{\frac{1}{2}} \cdot r^{\frac{3}{2}}$. If the constant for a fictional star system is $k = 4 \cdot 10^{-6}$ and a planet orbits at a mean radius of $r = 25 \cdot 10^4$ km, calculate the exact orbital period using the properties of powers with rational exponents and radicals.

In maritime navigation, the distance $d$ (in kilometres) to the visible horizon for an observer situated at a height $h$ (in metres) above sea level is approximated by the formula $d = \sqrt{13h}$. Two observers are at different points on a cliff at heights of $h_1 = 52$ m and $h_2 = 117$ m respectively. Calculate the exact difference between their distances to the visible horizon, simplifying the expression $\sqrt{13 \cdot 117} - \sqrt{13 \cdot 52}$ by taking out factors.