Question

1. \\(196 \div 3 \times 2^3 + (7 - 3)\\)

\\(196 \div 3 \times 2^3 + 4\\)
\\(67\\) \\(22\\)

Explanation:

Identify the expression

The handwritten image shows an expression being simplified step-by-step. Let's transcribe the initial expression:

$$ 198 \div [3 \times 2^2 + (7 - 3)] $$

Note: The handwriting uses angle brackets/braces and square brackets, which represent standard grouping symbols.

Evaluate parentheses

First, we simplify the innermost parentheses:

$$ 7 - 3 = 4 $$

This matches the first simplification step shown in the image:

$$ 198 \div [3 \times 2^2 + 4] $$

Evaluate exponents

Next, we evaluate the exponent inside the brackets:

$$ 2^2 = 4 $$

So the expression inside the brackets becomes:

$$ 3 \times 4 + 4 $$

Evaluate multiplication and addition

Now, perform multiplication before addition inside the brackets:

$$ 3 \times 4 = 12 $$

Then add \(4\):

$$ 12 + 4 = 16 $$

This is the value of the entire bracketed expression.
(Note: The student's work in the image shows a value of \(22\) under \(3 \times 2^2 + 4\), which is incorrect because \(3 \times 4 + 4 = 16\), not \(22\). It seems they calculated \(3 \times 2 = 6\), then \(6^2 = 36\), or made another arithmetic error).

Perform final division

Finally, divide \(198\) by the simplified bracket value:

$$ 198 \div 16 = 12.375 $$

If we look at the student's handwritten steps, they wrote:

$$ 198 \div [3 \times 2^2 + (7 - 3)] $$

$$ 198 \div [3 \times 2^2 + 4] $$

Then they grouped \(3 \times 2^2 + 4\) and wrote \(22\) (which is incorrect).
Then they grouped \(198 \div 22\) and wrote \(67\) (which is also incorrect, as \(198 \div 22 = 9\)).
Let's provide the correct step-by-step evaluation of the expression.

Answer:

The correct evaluation of the expression \(198 \div [3 \times 2^2 + (7 - 3)]\) is <blank>\(12.375\)</blank> (or \(\frac{99}{8}\)).