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) $\sqrt[6]{125}$

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

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) $x^{1,5}$

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:

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

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

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

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

• 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.

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

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

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:

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

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 $A = \frac{\sqrt{3}}{4}a^2$)

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

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