To evaluate [tex]\(\left(-3 \frac{2}{3}\right)^2\)[/tex], follow these steps:
### Step 1: Convert the Mixed Number to an Improper Fraction
First, let's convert the mixed number [tex]\(-3 \frac{2}{3}\)[/tex] to an improper fraction.
To do this:
- Multiply the whole number by the denominator of the fraction part: [tex]\( 3 \times 3 = 9 \)[/tex].
- Add the numerator of the fraction part to this product: [tex]\( 9 + 2 = 11 \)[/tex].
- Place this sum over the original denominator: [tex]\(-3 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{11}{3}\)[/tex].
The mixed number [tex]\(-3 \frac{2}{3}\)[/tex] as an improper fraction is [tex]\(-\frac{11}{3}\)[/tex].
### Step 2: Square the Improper Fraction
Now, we need to square the improper fraction [tex]\(-\frac{11}{3}\)[/tex].
Using the formula [tex]\(\left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2}\)[/tex]:
- Square the numerator: [tex]\( (-11)^2 = 121 \)[/tex].
- Square the denominator: [tex]\( 3^2 = 9 \)[/tex].
Thus, [tex]\(\left( -\frac{11}{3} \right)^2 = \frac{121}{9}\)[/tex].
### Step 3: Convert the Improper Fraction to a Decimal
To obtain the decimal form of [tex]\(\frac{121}{9}\)[/tex], perform the division:
[tex]\[
\frac{121}{9} \approx 13.444444444444443
\][/tex]
The numerical value of [tex]\(\left(-3 \frac{2}{3}\right)^2\)[/tex] is approximately [tex]\( 13.444444444444443 \)[/tex].
