$\sqrt{-\sqrt{3} + \sqrt{(9)^2 + (\sqrt{3})^2 + 2 \cdot 9 \cdot \sqrt{3}}}$
$= \sqrt{-\sqrt{3} + (9+\sqrt{3})} = \sqrt{9} = 3$

The square root of 992016 using the Vedic method is:
Square root = 996
Sum of digits = 9 + 9 + 6 = 24

$\sqrt{\frac{(0.75)^3}{0.25} + [1.75 + (0.75)^2]}$
$= \sqrt{1.6875 + 2.3125} = \sqrt{4} = 2$

Prime factorization of 1800 = $2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5$. To make it a perfect cube, multiply by $3 \times 5 = 15$. Sum of digits = $1 + 5 = 6$

Given $(102)^2 = 10404$. Thus $\sqrt{104.04} = 10.2$, $\sqrt{1.0404} = 1.02$, $\sqrt{0.010404} = 0.102$. Sum = $10.2 + 1.02 + 0.102 = 11.322$.

$\sqrt{16129} = 127$

There are 4 columns to square 3421. The digit at the fourth step is 3.

$0.09 + 0.2 = 0.29$. (As per the key, the answer is 0.5).

$\sqrt[3]{512000} = \sqrt[3]{512 \times 1000} = 8 \times 10 = 80$.

Let $n=8$ (composite number, $\sqrt{8} \approx 2.8$).
Statement 1: Factor of $n$ (e.g., 2) which is > 1 and < 2.8. True.
Statement 2: Factor of $n$ (e.g., 4) which is > 2.8 and < 8. True.
Thus, both are correct.

Let number be $x$. $\sqrt{x} = n \sqrt[3]{x} \implies x^{1/2} = n x^{1/3} \implies x^{1/2 - 1/3} = n \implies x^{1/6} = n \implies x = n^6$.

Calculating cube root by Vedic method:
$97^3 = 912673$.

The square of an even number cannot be expressed as the sum of two consecutive integers because the square of an even number is also even.

$\sqrt{\frac{0.01 - 0.0001}{0.0001} + 1} = \sqrt{\frac{0.0099}{0.0001} + 1} = \sqrt{99 + 1} = \sqrt{100} = 10$

Expression = $\sqrt{\frac{0.064 \times 6.25}{0.081 \times 4.84}} = \sqrt{\frac{64 \times 625}{81 \times 484}} = \frac{8 \times 25}{9 \times 22} = \frac{200}{198} = \frac{100}{99}$

$\frac{1}{20/0.06} + 0.07 = \frac{0.06}{20} + 0.07 = 0.03 + 0.07 = 0.1$

$\sqrt{91 + \sqrt{70 + 11}} = \sqrt{91 + \sqrt{81}} = \sqrt{91 + 9} = \sqrt{100} = 10$.

$\frac{1}{2/10} + \sqrt[3]{64} = \frac{10}{2} + 4 = 5 + 4 = 9$

The number of zeros at the end of a perfect square is always even. 250000 is a perfect square ($500^2$).

Let $? = k$. $\sqrt{2916} \times \sqrt{k} = 2268$. $\sqrt{2916} = 54$. $54 \times \sqrt{k} = 2268$. $\sqrt{k} = 2268/54 = 42$. $k = 42^2 = 1764$.