91Ó°ÊÓ

Solve the given problems. One leg of a right triangle is \(2 \sqrt{7}\) and the hypotenuse is \(6 .\) What is the area of the triangle?

Perform the required operation. Change to radicals of the same order: \(3 x^{2 / 3} ; 2 y^{1 / 2} ;(5 z)^{1 / 4}\)

Perform the indicated operations. A factor used in measuring the loudness sensed by the human ear is \(\left(I / I_{0}\right)^{0.3},\) where \(I\) is the intensity of the sound and \(I_{0}\) is a reference intensity. Evaluate this factor for \(I=3.2 \times 10^{-6} \mathrm{W} / \mathrm{m}^{2}\) (ordinary conversation) and \(I_{0}=10^{-12} \mathrm{W} / \mathrm{m}^{2}\).

In Exercises \(53-76,\) solve the given problems. Solve for \(x: 3^{-1} x^{2}=-9 x^{-1}\)

Solve the given problems.Solve for $$x: 3^{-1} x^{2}=-9 x^{-1}$$.

Perform the required operation. Display the graphs of \(y_{1}=\sqrt{x+2}\) and \(y_{2}=\sqrt{x}+\sqrt{2}\) on a calculator to show that \(\sqrt{x+2}\) is not equal to \(\sqrt{x}+\sqrt{2}\).

Solve the given problems. Solve for $$x: 2^{5 x}=2^{7}\left(2^{2 x}\right)^{2}$$.

Solve the given problems. For an object oscillating at the end of a spring and on which there is a force that retards the motion, the equation \(m^{2}+b m+k^{2}=0\) must be solved. Here, \(b\) is a constant related to the retarding force, and \(k\) is the spring constant. By substitution, show that \(m=\frac{1}{2}(\sqrt{b^{2}-4 k^{2}}-b)\) is a solution.

Perform the indicated operations. The period \(T\) of a satellite circling earth is given by \(T^{2}=k R^{3}\left(1+\frac{d}{R}\right)^{3},\) where \(R\) is the radius of earth, \(d\) is the distance of the satellite above earth, and \(k\) is a constant. Solve for \(R,\) using fractional exponents in the result.

In Exercises \(53-76,\) solve the given problems. Solve for \(x: 2^{5 x}=2^{7}\left(2^{2 x}\right)^{2}\)