Answer :
Certainly! Let's break down the expression step-by-step and simplify it to obtain the result.
First, we start with the expression:
[tex]\[ \left(\frac{\mid 2 r^2 t 1^3}{4 t^2}\right)^2 \][/tex]
### Step 1: Simplify Inside the Absolute Value
Look inside the absolute value bars [tex]\(\mid \cdot \mid\)[/tex]. The term [tex]\(1^3\)[/tex] simplifies to [tex]\(1\)[/tex]. Any term raised to the power of three (or any power) that is equal to 1 remains 1. Multiplying by 1 does not change the value of the expression. So, remove [tex]\(1^3\)[/tex] from the numerator:
[tex]\[ \left(\frac{\mid 2 r^2 t \cdot 1 \mid }{4 t^2} \right)^2 \quad \Rightarrow \quad \left( \frac{\mid 2 r^2 t \mid}{4 t^2} \right)^2 \][/tex]
### Step 2: Simplify the Fraction Inside the Absolute Value
Now, simplify the fraction inside the absolute value.
[tex]\[ \frac{\mid 2 r^2 t \mid }{4 t^2} \][/tex]
The [tex]\(t\)[/tex] term in the numerator and [tex]\(t^2\)[/tex] term in the denominator can be reduced by cancelling one [tex]\(t\)[/tex]:
[tex]\[ \frac{\mid 2 r^2 t \mid}{4 t^2} \quad \Rightarrow \quad \frac{\mid 2 r^2 \mid}{4 t} \][/tex]
Since absolute values maintain only non-negative values, the absolute value doesn't change because both [tex]\(2\)[/tex] and [tex]\(r^2\)[/tex] are already non-negative (noting [tex]\(r\)[/tex] is squared):
[tex]\[ \frac{2 r^2}{4 t} \][/tex]
### Step 3: Further Simplification of the Fraction
Next, simplify [tex]\( \frac{2 r^2}{4 t} \)[/tex]:
[tex]\[ \frac{2 r^2}{4 t} = \frac{2}{4} \cdot \frac{r^2}{t} = \frac{1}{2} \cdot \frac{r^2}{t} = \frac{r^2}{2 t} \][/tex]
### Step 4: Square the Result
Finally, we take the simplified expression and square it as indicated in the original problem:
[tex]\[ \left( \frac{r^2}{2 t} \right)^2 = \frac{(r^2)^2}{(2 t)^2} = \frac{r^4}{4 t^2} \][/tex]
### Result
Thus, the simplified form of the given expression
[tex]\[ \left(\frac{\mid 2 r^2 t 1^3 \mid}{4 t^2}\right)^2 \][/tex]
is
[tex]\[ \frac{r^4}{4 t^2} \][/tex]
First, we start with the expression:
[tex]\[ \left(\frac{\mid 2 r^2 t 1^3}{4 t^2}\right)^2 \][/tex]
### Step 1: Simplify Inside the Absolute Value
Look inside the absolute value bars [tex]\(\mid \cdot \mid\)[/tex]. The term [tex]\(1^3\)[/tex] simplifies to [tex]\(1\)[/tex]. Any term raised to the power of three (or any power) that is equal to 1 remains 1. Multiplying by 1 does not change the value of the expression. So, remove [tex]\(1^3\)[/tex] from the numerator:
[tex]\[ \left(\frac{\mid 2 r^2 t \cdot 1 \mid }{4 t^2} \right)^2 \quad \Rightarrow \quad \left( \frac{\mid 2 r^2 t \mid}{4 t^2} \right)^2 \][/tex]
### Step 2: Simplify the Fraction Inside the Absolute Value
Now, simplify the fraction inside the absolute value.
[tex]\[ \frac{\mid 2 r^2 t \mid }{4 t^2} \][/tex]
The [tex]\(t\)[/tex] term in the numerator and [tex]\(t^2\)[/tex] term in the denominator can be reduced by cancelling one [tex]\(t\)[/tex]:
[tex]\[ \frac{\mid 2 r^2 t \mid}{4 t^2} \quad \Rightarrow \quad \frac{\mid 2 r^2 \mid}{4 t} \][/tex]
Since absolute values maintain only non-negative values, the absolute value doesn't change because both [tex]\(2\)[/tex] and [tex]\(r^2\)[/tex] are already non-negative (noting [tex]\(r\)[/tex] is squared):
[tex]\[ \frac{2 r^2}{4 t} \][/tex]
### Step 3: Further Simplification of the Fraction
Next, simplify [tex]\( \frac{2 r^2}{4 t} \)[/tex]:
[tex]\[ \frac{2 r^2}{4 t} = \frac{2}{4} \cdot \frac{r^2}{t} = \frac{1}{2} \cdot \frac{r^2}{t} = \frac{r^2}{2 t} \][/tex]
### Step 4: Square the Result
Finally, we take the simplified expression and square it as indicated in the original problem:
[tex]\[ \left( \frac{r^2}{2 t} \right)^2 = \frac{(r^2)^2}{(2 t)^2} = \frac{r^4}{4 t^2} \][/tex]
### Result
Thus, the simplified form of the given expression
[tex]\[ \left(\frac{\mid 2 r^2 t 1^3 \mid}{4 t^2}\right)^2 \][/tex]
is
[tex]\[ \frac{r^4}{4 t^2} \][/tex]
