Question

9 =
10 =
11 =
12 =
13 =
14 =
15 =
16 =
17 =
18 =
19 =
20 =
21 =
22 =
23 =
24 =
25 =
168

Explanation:

Step1: Analyze the goal (find expression for 19 using 4s)

We need to create an expression with four 4s (or operations involving 4, like factorial, square root, exponent) that equals 19. Let's recall operations: factorial (\(4! = 24\)), square root (\(\sqrt{4}=2\)), exponent (\(4^0 = 1\)), addition, subtraction, multiplication, division.

Step2: Brainstorm possible combinations

Let's start with factorial: \(4! = 24\). We need to reduce 24 to 19. \(24 - 5 = 19\), but how to get 5 from three 4s? Wait, maybe use \(4 + \sqrt{4} + 4^0\)? Wait, no, we have four 4s. Wait, \(4! - (4 + \sqrt{4} + 4^0)\)? Wait, \(4! = 24\), \(4 + \sqrt{4} + 4^0 = 4 + 2 + 1 = 7\), \(24 - 7 = 17\). Not 19. Wait, \(4 \times 4 + \sqrt{4} + 4^0\)? \(16 + 2 + 1 = 19\)! Wait, \(4 \times 4 = 16\), \(\sqrt{4}=2\), \(4^0 = 1\). So \(4 \times 4 + \sqrt{4} + 4^0\). Let's check: \(16 + 2 + 1 = 19\). Yes! That uses four 4s (4, 4, \(\sqrt{4}\) (which is 4 under root), and \(4^0\) (base 4)).

Step3: Verify the expression

• \(4 \times 4 = 16\)
• \(\sqrt{4} = 2\)
• \(4^0 = 1\)
• Sum: \(16 + 2 + 1 = 19\)

So the expression is \(4 \times 4 + \sqrt{4} + 4^0\) (or other equivalent, but this works with four 4s: two 4s for multiplication, one 4 for square root, one 4 for exponent).

Answer:

\(4 \times 4 + \sqrt{4} + 4^0\) (or equivalent, e.g., \(4\times4+\sqrt{4}+1\) but using 4s for the 1 as \(4^0\))