Powers with Rational Exponents. Radicals

In the study of microorganisms, the size of certain structures can be expressed using powers and roots. Convert the following radicals into powers with fractional exponents and vice versa:

1. a) $\sqrt[5]{x^2}$

1. b) $\frac{1}{\sqrt[3]{a}}$

1. c) $7^{\frac{3}{4}}$

1. d) $x^{-\frac{1}{2}}$

Simplify the following radicals by dividing the index and the exponent of the radicand by their highest common factor:

1. a) $\sqrt[12]{x^8}$

1. b) $\sqrt[9]{27}$

1. c) $\sqrt[10]{a^{15}}$

1. d) $\sqrt[6]{125}$

Express the following powers with rational exponents in radical form and simplify the result by taking out as many factors as possible:

1. a) $27^{\frac{2}{3}}$

1. b) $125^{-\frac{1}{3}}$

1. c) $\left(\frac{16}{81}\right)^{\frac{3}{4}}$

1. d) $x^{1.5}$

To optimise space in a graphic design, a programmer needs to extract factors from the radicals whenever possible. Extract factors from the radicals in the following expressions:

1. a) $\sqrt{50}$

1. b) $\sqrt[3]{16x^5}$

1. c) $\sqrt{a^3 b^4 c}$

1. d) $\sqrt[4]{32}$

In some financial formulas, it is preferable to have all elements under a single radical sign. Insert the factors inside the root and simplify the result:

1. a) $3\sqrt{2}$

1. b) $x^2 \sqrt[3]{x}$

1. c) $\frac{2}{3}\sqrt{5}$

1. d) $a \sqrt[4]{a^3}$

An interior designer has planned a square rug for a law firm in Mexico City. The total area of the rug is $18 \text{ m}^2$.

1. a) Express the length of the side of the square exactly using a radical.
2. b) Simplify the radical by extracting factors.

An architect is calculating the perimeters of a metal structure made up of several sections. Add and simplify the following like radicals to obtain the total length:

$2\sqrt{27} - 5\sqrt{3} + \sqrt{12}$

An architectural firm is designing a triangular garden. The lengths of the three paths bordering the garden are given by the following expressions in meters:

• Path A: $3\sqrt{20}$

• Path B: $\sqrt{125}$

• Path C: $2\sqrt{45}$

Calculate the total perimeter of the garden, simplifying the expression as far as possible.

Find the value of $x$ that satisfies the following equation, simplifying all radicals beforehand:

$\displaystyle \sqrt{125x} - 2\sqrt{45x} + \sqrt{500x} = 18\sqrt{5}$

Perform the following multiplication and division operations with radicals having the same index, simplifying the final result:

1. a) $\sqrt[3]{2} \times \sqrt[3]{4}$

1. b) $\frac{\sqrt{18}}{\sqrt{2}}$

1. c) $\sqrt[5]{x^2} \times \sqrt[5]{x^3}$

1. d) $(\sqrt[4]{a})^2$

Calculate and simplify the following expression, leaving the result as a single radical with the smallest possible index:

$$\frac{\sqrt[3]{a^2} \times \sqrt[4]{a^3}}{\sqrt[6]{a^5}}$$

Solve the following combined operation with radicals, expressing the result in the simplest possible form:

$$\left( \sqrt{12} - \sqrt{3} \right) \times \sqrt[3]{9}$$

Hint: First express all terms as powers with base 3.

Simplify the following expression by reducing to a single radical and taking out factors if necessary:

$$\sqrt{x \times \sqrt[3]{x^2 \times \sqrt{x}}}$$

In an entrance exam, students are asked to determine which of the following two metal pieces has a greater area. Piece A is a square with side length $L_A = \sqrt[3]{4}$ cm and piece B is a square with side length $L_B = \sqrt[4]{6}$ cm.

Calculate the area of each piece and compare them by reducing the radicals to a common index.

In the design of a new microelectronic component, its resistance is given by the expression:

$\displaystyle R = \sqrt[4]{27} \times \sqrt[3]{9} \times \sqrt{3}$

Express the radicals as powers of base 3 to find the value of $R$ as a single power with a fractional exponent. Afterwards, state which two consecutive integers this value lies between.

The volume of a cubic container used to transport tech goods is $V = 16\sqrt{2}$ cubic meters.

1. a) Express the edge of the cube as a power with base 2.
2. b) Calculate the exact length of the edge in simplified radical form.

Simplify the following algebraic expression by operating with the radicals and taking out as many factors as possible. Express the final result using a single radical:

$\displaystyle \frac{\sqrt{x^3 \times \sqrt[3]{x^2 \times y}}}{\sqrt[6]{x \times y^5}} \times \left( \frac{x}{y^2} \right)^{-1/2}$

Simplify the following combined expression, leaving the result in its simplest form:

$\displaystyle \frac{\left( \sqrt[3]{a^2 \times b} \right)^2 \times \sqrt{a \times b^3}}{\sqrt[6]{a^5 \times b}}$

Calculate the exact total surface area of a regular octahedron (formed by two square-based pyramids joined at their base) whose edge length is $\sqrt{6} + \sqrt{2}$ cm. You must leave the result simplified in radical form, without using decimal approximations.

(Hint: The area of an equilateral triangle of side length $a$ is $A = \frac{\sqrt{3}}{4}a^2$)

An architect is designing a right triangular window. The hypotenuse measures $\sqrt[4]{1024}$ meters and one of the shorter sides measures $\sqrt{2}$ meters.

1. a) Calculate the exact length of the other side in simplified radical form.
2. b) Calculate the area of the window. Is the area a rational or irrational number? Justify your answer.

A cubic water tank is to be constructed with an interior diagonal measuring exactly $\sqrt{108}$ meters.

1. a) Show that the edge length of the tank is a rational number.
2. b) If the cost of waterproofing the interior walls (the 4 side faces and the floor) is $3\,000 \times \sqrt{3}$ pesos per square meter, calculate the exact total cost of the work.

Rationalise the following expressions to eliminate the radicals from the denominator, simplifying the result where possible:

1. a) $\frac{5}{\sqrt{3}}$

1. b) $\frac{1}{\sqrt[3]{2}}$

1. c) $\frac{\sqrt{2}}{\sqrt{5}}$

1. d) $\frac{x}{\sqrt{x}}$

Rationalise and simplify the following expressions so that they do not contain roots in the denominator:

1. a) $\frac{10}{\sqrt[3]{5^2}}$

1. b) $\frac{2\sqrt{3}}{\sqrt{7} - \sqrt{2}}$

1. c) $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}$

Rationalise and simplify the following expression as far as possible. Ensure that the final denominator is an integer:

$\displaystyle \frac{2\sqrt{3}}{\sqrt{2} + \sqrt{3} - \sqrt{5}}$