Rational Exponents and Radicals

Express the following powers with fractional exponents as a radical 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 of the exponent:

• a) $\sqrt[6]{x^2}$

• b) $\sqrt[15]{a^5}$

• c) $\sqrt[8]{2^6}$

Extract all possible factors from inside the following radicals:

• a) $\sqrt{50}$

• b) $\sqrt[3]{16}$

• c) $\sqrt[4]{x^7 y^5}$

Introduce the factors inside 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 $a \ge 0$

Perform the following products of radicals with the same index and extract 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, introducing the exponent into the radicand and expressing the result in the simplest possible form:

• a) $(\sqrt[3]{5})^6$

• b) $(\sqrt{x^3})^4$

A landscape architect is designing a green area in the shape of a scalene triangle and needs to calculate its perimeter to order a fence. The plans indicate that the sides measure $\sqrt{48}$ ft, $\sqrt{75}$ ft, and $\sqrt{12}$ ft. Simplify the radicals by extracting factors and calculate the exact perimeter expressed as a single radical.

An agricultural cooperative wants to fence a square-shaped plot of land whose total area is $1250 \text{ ft}^2$. Calculate the exact length of the side of the land by extracting factors from the radical and determine how many feet of fence they will need in total, leaving the result in its simplest radical form.

Express as a single radical using the root of a root property 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 inches, are given by the expressions $A = \sqrt[3]{5}$, $B = \sqrt[4]{7}$ and $C = \sqrt[6]{24}$. Reduce the radicals to a common index to order the thicknesses of the sheets from smallest to largest.

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

Perform the following operations by reducing to a common index first and extract the possible factors from the final result:

• a) $\sqrt{2} \cdot \sqrt[3]{4} \cdot \sqrt[4]{8}$

• b) $\frac{\sqrt[4]{27}}{\sqrt[6]{3}}$

Operate using the properties of fractional powers and express the final result as a simplified root:

Simplify the following algebraic expression fully, assuming that $a$ and $b$ are positive real numbers. Express the final result using radical notation, without leaving negative or fractional exponents in the solution:

Find the value of $x$ that satisfies the following equation, first transforming all expressions into powers of base $2$ with rational exponents:

Express the final solution as an irreducible fraction.

Prove, rigorously using the properties of powers with rational exponents, that for any positive real number $x$, the expression $\sqrt[n]{\sqrt[m]{x}}$ is equivalent to $\sqrt[m]{\sqrt[n]{x}}$. Then, apply this property or convert to powers to calculate the exact numerical value of the following sum:

Simplify the following expression with nested radicals, reducing it to a single common index and extracting any possible factors outside the resulting root:

In a biology laboratory, the number of bacteria in a certain 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 there will be when $2$ hours have elapsed.

The main diagonal of a cubic water storage tank (the straight line joining two opposite vertices passing through the center of the cube) measures exactly $\sqrt{108}$ feet. Calculate the exact edge length of the cube and, from it, determine the capacity of the tank in cubic feet. Express all results by simplifying the radicals.

The final capital $C_f$ obtained by investing an initial capital $C_0$ at an annual compound interest rate $r$ (expressed as a decimal) for $t$ years is given by the formula $C_f = C_0 (1+r)^t$. A company invests $\$10000$ at an interest rate of $21$ % per year ($r = 0,21$). Using powers with fractional exponents, calculate exactly the accumulated capital after $1$ year and $4$ months (which is equivalent to $t = \frac{4}{3}$ of a year). Knowing that $1,21 = 1,1^2$, simplify the result by extracting the possible factors from the radical.

The period $T$ (in seconds) that a pendulum takes to complete one full oscillation is calculated with the physical formula $T = 2\pi \sqrt{\frac{L}{g}}$, where $L$ is the length of the string in feet and $g$ is the acceleration due to gravity (we will take $g = 10 \text{ ft/s}^2$). If a pendulum in a museum has a length of $L = 12$ ft, calculate its exact period by simplifying the radical fully and leaving it in terms of $\pi$.

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 for a fictional star system the constant is $k = 4 \cdot 10^{-6}$ and a planet orbits at a mean radius of $r = 25 \cdot 10^4$ miles, calculate the exact orbital period using the properties of powers with rational exponents and radicals.

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