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) $\sqrt[8]{2^6}$

Calculate using the laws of fractional exponents and express the final result as a simplified radical:

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.

Fully simplify the following algebraic expression, assuming that $a$ and $b$ are positive real numbers. Express the final result using radical notation, leaving no negative or fractional exponents in the solution:

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 $a \ge 0$

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

In maritime navigation, the distance $d$ (in kilometers) to the visible horizon for an observer situated at a height $h$ (in meters) 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 $\sqrt{13 \times 117} - \sqrt{13 \times 52}$ by taking out factors.

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

• a) $\sqrt{3} \times \sqrt{12}$

• b) $\sqrt[3]{4} \times \sqrt[3]{10}$

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 millimeters, 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.

Perform the following operations, first expressing the radicals with a common index, and take out any possible factors from the final result:

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

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

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

Simplify the following expression with nested radicals, reducing it to a single common index and taking out as many factors as possible from the resulting radical:

Prove, using strictly 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 conversion to powers to calculate the exact numerical value of the following sum:

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.

An agricultural cooperative wants to fence a square piece of land whose total area is $1250 \text{ m}^2$. Calculate the exact length of the side of the land by taking out factors from the radical and determine how many meters of fencing they will need in total, leaving the result in its simplest radical form.

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

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

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

The final amount $C_f$ obtained by investing an initial principal $C_0$ 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 $200\,000$ pesos at an interest rate of $21$ % per year ($r = 0.21$). Using powers with fractional exponents, calculate the exact accumulated amount 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 taking out as many factors as possible from the radical.