91Ó°ÊÓ

The following problems involve addition, subtraction, and multiplication of radical expressions, as well as rationalizing the denominator. Perform the operations and simplify, if possible. All variables represent positive real numbers. $$ \sqrt{y^{2}(x+y)} \sqrt{(x+y)^{3}} $$

Use a calculator to solve each problem. Round answers to the nearest tenth. Embroidery. The radius \(r\) of a circle is given by the formula \(r=\sqrt{\frac{A}{\pi}},\) where \(A\) is its area. Find the diameter of the embroidery hoop if there are 38.5 in. \(^{2}\) of stretched fabric on which to embroider.

The frequency of vibration of a string varies directly as the square root of the tension and inversely as the length of the string. Suppose a string 2.5 feet long, under a tension of 16 pounds, vibrates 25 times per second. Find \(k\) the constant of proportionality.

Simplify each expression. All variables represent positive real numbers. $$ \sqrt[4]{25 b^{2}} $$

Simplify: \(\left(i^{349}\right)^{-i^{456}}\)

Explain why \(\sqrt{36}\) is just \(6,\) and not also \(-6\).

Logging. The width \(w\) and height \(h\) of the strongest rectangular beam that can be cut from a cylindrical log of radius \(a\) are given by \(w=\frac{2 a}{3}\left(3^{1 / 2}\right)\) and \(h=a\left(\frac{8}{3}\right)^{1 / 2} .\) Find the width, height, and cross-sectional area of the strongest beam that can be cut from a log with diameter 4 feet. Round to the nearest hundredth.

The fraction \(\frac{2}{4}\) is equal to \(\frac{1}{2} .\) Is \(16^{2 / 4}\) equal to \(16^{1 / 2} ?\) Explain.