Problem 127 - Intermediate Algebra

Question

In physics, the speed of a wave traveling over a stretched string with tension \(t\) and density \(u\) is given by the expression \(\frac{\sqrt{t}}{\sqrt{u}} .\) Write this expression with rational exponents.

Step-by-Step Solution

Answer

\( t^{1/2} u^{-1/2} \)

Step 1: Understanding the Expression

The speed of a wave on a stretched string is given by \( \frac{\sqrt{t}}{\sqrt{u}} \). This expression uses square root notations for both \( t \) and \( u \).

Step 2: Converting Square Roots to Exponents

Recall that the square root \( \sqrt{x} \) can be expressed as \( x^{1/2} \). Therefore, \( \sqrt{t} \) can be written as \( t^{1/2} \) and \( \sqrt{u} \) as \( u^{1/2} \).

Step 3: Expressing the Division with Exponents

The division of two expressions with exponents can be written using subtraction of the exponents' powers: \( \frac{a^m}{b^n} = a^m b^{-n} \). Thus, \( \frac{t^{1/2}}{u^{1/2}} = t^{1/2} u^{-1/2} \).

Step 4: Final Expression Using Rational Exponents

Combine the properties from earlier steps: the expression for the speed of the wave using rational exponents is \( t^{1/2} u^{-1/2} \).

Key Concepts

Square Roots

Square roots are a fundamental concept often represented by the radical symbol \( \sqrt{} \). They are used to determine what number, when multiplied by itself, gives the original number. For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \). In various mathematical expressions, especially in physics and engineering, square roots often need to be converted to exponents for simplification and calculation purposes. In mathematics, a square root can be transformed into an exponent of \( \frac{1}{2} \). Thus, the expression \( \sqrt{x} \) can equivalently be written as \( x^{1/2} \).

This conversion is particularly useful when working with equations involving multiple exponents or when applying mathematical operations like multiplication and division. By expressing square roots as rational exponents, it becomes easier to manipulate the equations using exponent rules. When dealing with the physics of waves on a string, like in our original exercise, transforming the square roots of tension and density into rational exponents allows us to apply more advanced mathematical strategies efficiently.

Exponent Rules

Understanding exponent rules is crucial for simplifying expressions involving powers and roots. An exponent indicates how many times a number—known as the base—is multiplied by itself. For instance, in \( x^3 \), \( x \) is the base and \( 3 \) is the exponent, meaning \( x \times x \times x \).

When dealing with powers of the same base, several key rules come into play:

• Product of Powers: \( x^m \times x^n = x^{m+n} \)
• Quotient of Powers: \( \frac{x^m}{x^n} = x^{m-n} \)
• Power of a Power: \( (x^m)^n = x^{mn} \)

These rules simplify the manipulation of complex expressions. For instance, when you encounter \( \frac{t^{1/2}}{u^{1/2}} \) as in the exercise, using the quotient of powers rule helps rewrite the expression as \( t^{1/2} u^{-1/2} \). The negative exponent here indicates division by that base, providing a more streamlined way to handle these expressions in both mathematics and physics.

Physics Applications

Rational exponents and square roots take on significant importance in the realm of physics, especially when deriving formulas for wave behavior on materials like strings. In physics, expressions often need to represent relationships more directly or reveal underlying principles through mathematical reductions.

When you have a wave traveling over a stretched string, the speed is given by \( \frac{\sqrt{t}}{\sqrt{u}} \), where \( t \) is the tension and \( u \) is the density. By converting the equation into rational exponents \( t^{1/2} u^{-1/2} \), physicists can analyze and comprehend how changes in variables affect the speed. This conversion is more than just a mathematical exercise—it provides deeper insight into how wave properties are influenced by the physical parameters of the medium they're moving through.

Such principles extend beyond strings to encompass other physical systems where wave phenomena are central, such as acoustics, optics, and quantum mechanics. Simplified expressions help scientists and engineers to predict system behaviors efficiently and design solutions to complex engineering problems.