To find the value of the expression [tex]\( t \sqrt{\frac{1-a}{1+a}} \)[/tex], let's break it down step-by-step.
### Step 1: Understand the Expression
We need to evaluate the expression [tex]\( t \sqrt{\frac{1-a}{1+a}} \)[/tex].
### Step 2: Analyze the Components
- [tex]\( t \)[/tex] is a scalar multiplier.
- [tex]\( \sqrt{\frac{1-a}{1+a}} \)[/tex] is a square root function applied to the fraction [tex]\(\frac{1-a}{1+a}\)[/tex].
### Step 3: Evaluate the Inner Fraction
First, compute the fraction [tex]\(\frac{1-a}{1+a}\)[/tex].
### Step 4: Apply the Square Root
Next, take the square root of the fraction computed in Step 3.
### Step 5: Multiply by Scalar [tex]\( t \)[/tex]
Finally, multiply the result from Step 4 with the scalar [tex]\( t \)[/tex].
### Detailed Solution
1. Compute the Fraction:
[tex]\[
\frac{1-a}{1+a}
\][/tex]
2. Apply the Square Root:
[tex]\[
\sqrt{\frac{1-a}{1+a}}
\][/tex]
3. Multiply by [tex]\( t \)[/tex]:
[tex]\[
t \times \sqrt{\frac{1-a}{1+a}}
\][/tex]
Let’s illustrate this with an example. Suppose [tex]\( t = 2 \)[/tex] and [tex]\( a = 0.5 \)[/tex]:
Step 1: Compute the Fraction:
[tex]\[
\frac{1 - 0.5}{1 + 0.5} = \frac{0.5}{1.5} = \frac{1}{3}
\][/tex]
Step 2: Apply the Square Root:
[tex]\[
\sqrt{\frac{1}{3}} = \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
\][/tex]
Step 3: Multiply by [tex]\( t \)[/tex]:
[tex]\[
2 \times \frac{\sqrt{3}}{3} = \frac{2\sqrt{3}}{3}
\][/tex]
Thus, the value of the expression when [tex]\( t = 2 \)[/tex] and [tex]\( a = 0.5 \)[/tex] is [tex]\( \frac{2\sqrt{3}}{3} \)[/tex].
### General Expression
Thus, in general, the value of the expression [tex]\( t \sqrt{\frac{1-a}{1+a}} \)[/tex] depends on the specific values of [tex]\( t \)[/tex] and [tex]\( a \)[/tex]. By following the steps outlined, you can substitute any given values for [tex]\( t \)[/tex] and [tex]\( a \)[/tex] to find the result.
