The problem consists of three parts: (a) Evaluate and express the answer in standard form: $\frac{4.56 \times 3.6}{0.12}$ (b) Evaluate without using mathematical tables or calculator: $(73.8)^2 - (26.2)^2$ (c) Simplify $\sqrt{1\frac{19}{81}}$, expressing the answer in the form $\frac{a}{b}$ where $a$ and $b$ are positive integers.

1. Problem Description

The problem consists of three parts:

(a) Evaluate and express the answer in standard form: $\frac{4.56 \times 3.6}{0.12}$

(b) Evaluate without using mathematical tables or calculator: $(73.8)^2 - (26.2)^2$

(c) Simplify $\sqrt{1\frac{19}{81}}$, expressing the answer in the form $\frac{a}{b}$ where $a$ and $b$ are positive integers.

2. Solution Steps

(a) $\frac{4.56 \times 3.6}{0.12}$

First, multiply $4.56$ by $3.6$:

$4.56 \times 3.6 = 16.416$

Then, divide $16.416$ by $0.12$:

$\frac{16.416}{0.12} = \frac{1641.6}{1.2} = \frac{16416}{12} = 1368$

(b) $(73.8)^2 - (26.2)^2$

Using the difference of squares formula: $a^2 - b^2 = (a+b)(a-b)$

$(73.8)^2 - (26.2)^2 = (73.8 + 26.2)(73.8 - 26.2)$

$73.8 + 26.2 = 100$

$73.8 - 26.2 = 47.6$

$(100)(47.6) = 4760$

(c) $\sqrt{1\frac{19}{81}}$

First, convert the mixed number to an improper fraction:

$1\frac{19}{81} = \frac{1 \times 81 + 19}{81} = \frac{81 + 19}{81} = \frac{100}{81}$

Then, take the square root:

$\sqrt{\frac{100}{81}} = \frac{\sqrt{100}}{\sqrt{81}} = \frac{10}{9}$

3. Final Answer

(a) 136.8

(b) 4760

(c) $\frac{10}{9}$