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Chapter 8: Problem 16

In the following exercises, evaluate the rational expression for the given values. \(\frac{2 s^{2} t}{s^{2}-9 t^{2}}\) (a) \(s=4, t=1\) (b) \(s=-1, t=-1\) c) \(s=0, t=2\)

Short Answer

Expert verified
(a) \(\frac{32}{7}\), (b) \(\frac{1}{4}\), (c) \(0\).

Step by step solution

01

Understand the Rational Expression

The given rational expression is \(\frac{2 s^{2} t}{s^{2}-9 t^{2}}\). We need to evaluate this expression by substituting the given values for \(s\) and \(t\).
02

Substitute Values for Part (a)

For \(s=4\) and \(t=1\), substitute these values into the expression: \(\frac{2 (4)^{2} (1)}{(4)^{2}-9 (1)^{2}}\).
03

Simplify Part (a)

Calculate each term: \(\frac{2 \times 16 \times 1}{16 - 9 \times 1} = \frac{32}{7}\).
04

Substitute Values for Part (b)

For \(s=-1\) and \(t=-1\), substitute these values into the expression: \(\frac{2 (-1)^{2} (-1)}{(-1)^{2}-9 (-1)^{2}}\).
05

Simplify Part (b)

Calculate each term: \(\frac{2 \times 1 \times -1}{1 - 9 \times 1} = \frac{-2}{-8} = \frac{1}{4}\).
06

Substitute Values for Part (c)

For \(s=0\) and \(t=2\), substitute these values into the expression: \(\frac{2 (0)^{2} (2)}{(0)^{2}-9 (2)^{2}}\).
07

Simplify Part (c)

Calculate each term: \(\frac{2 \times 0 \times 2}{0 - 36} = \frac{0}{-36} = 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

substitution
Substitution involves replacing variables in an expression with given numerical values. In our example, we must substitute the variables \(s\) and \(t\) with provided values. For instance, in part (a), we replace \(s = 4\) and \(t = 1\). This substitution allows us to simplify the original rational expression step by step.
Substitution is simply about taking each variable, placing the given numbers in their positions, and following standard algebraic operations.
expressions evaluation
Evaluating expressions is the process of finding the numerical value of an algebraic expression after substitution. Once we substitute given values into the expression \(\frac{2 s^{2} t}{s^{2}-9 t^{2}}\), we need to carry out operations like squaring, multiplication, and division.
For example, in part (a), once we substitute \(s = 4\) and \(t = 1\), we get \(\frac{2 (4)^{2} (1)}{(4)^{2}-9 (1)^{2}} = \frac{32}{7}\). Evaluating expressions combines basic arithmetic with following the order of operations.
fraction simplification
Fraction simplification involves reducing a fraction to its simplest form. After substitution in our example, we often get a complex fraction that needs simplifying.
In part (b), substituting \(s = -1\) and \(t = -1\) gives \(\frac{2 (-1)^{2} (-1)}{(-1)^{2}-9 (-1)^{2}} = \frac{-2}{-8}\). Simplifying \-2/-8\ results in \(\frac{2}{8} = \frac{1}{4}\).
Always look for common factors in the numerator and denominator to simplify.
algebraic fractions
Algebraic fractions have variables in the numerator, denominator, or both. They're simplified similarly to numerical fractions, but require handling variables carefully.
For example, our given expression, \(\frac{2 s^{2} t}{s^{2}-9 t^{2}}\), is an algebraic fraction. The operations performed must adhere to algebra rules. Simplification could involve factoring expressions or finding common denominators if needed.
Understanding algebraic fractions is crucial because it mixes both fractional and algebraic techniques.

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