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

1. Problem Description

We are given three problems:

(a) Evaluate the expression $\frac{4.56 \times 3.6}{0.12}$.

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

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

2. Solution Steps

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

$\frac{4.56 \times 3.6}{0.12} = \frac{4.56 \times 3.6 \times 100}{0.12 \times 100} = \frac{4.56 \times 360}{12} = 4.56 \times 30$

$4.56 \times 30 = 4.56 \times 3 \times 10 = 13.68 \times 10 = 136.8$

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

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

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

$= (100)(47.6) = 4760$

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

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) $\frac{10}{9}$