Answer :

Certainly! Let's break down the problem step by step and find the solution.

1. Understanding [tex]\(\overline{33}\)[/tex]:
- The notation [tex]\( \overline{33} \)[/tex] represents the repeating decimal [tex]\( 0.\overline{33} \)[/tex], which can be written as [tex]\( 0.3333\ldots \)[/tex].

2. Understanding the square root:
- [tex]\( \sqrt{144} \)[/tex] represents the square root of 144. The positive square root of 144 is [tex]\( 12 \)[/tex].

3. Multiplying:
- First, we convert the repeating decimal [tex]\( 0.3333\ldots \)[/tex] into a fraction. In fact, [tex]\( 0.3333\ldots = \frac{1}{3} \)[/tex].
- Now we multiply [tex]\( 8 \times 0.3333\ldots \)[/tex] or [tex]\( 8 \times \frac{1}{3} \)[/tex]:
[tex]\[ 8 \times 0.3333\ldots = \frac{8}{3} \approx 2.64 \][/tex]

4. Combining results with the square root:
- We multiply [tex]\( 8 \)[/tex] by [tex]\( \sqrt{144} = 12 \)[/tex]:
[tex]\[ 8 \times 12 = 96 \][/tex]

5. Final results:
- The calculation of [tex]\( \frac{8}{3} = 2.64 \)[/tex].
- The calculation of [tex]\( 8 \times 12 = 96 \)[/tex].

The results can be expressed as:

- The fraction part multiplied: [tex]\( 2.64 \)[/tex].
- The final multiplication result: [tex]\( 96 \)[/tex].

Therefore, the answer to [tex]\( 8 . \overline{33} \times \sqrt{144} \)[/tex] results in:
[tex]\[ 96.0 \][/tex]

These steps outline a clear and detailed approach to solving the problem by breaking down each part of the operation!

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