Prime factorization: 1 Marks
Number of Triangles: 1 Mark
Sum of the area: 1 Mark
Steps: 1 Mark
Given that,
The area of a certain number of triangles is equal to the exponents of the prime factors of the number 1628 and each prime factor represents a triangle.
Prime factors of 1628 are:
$1628=2^2\times 3\times 7\times 19$.
Since there are 5 prime factors, 
$\Rightarrow$ The number of given triangles are = 5
The area of the triangles is the sum of powers of the prime factors.
$\Rightarrow$The sum of areas of the triangle = 2 + 1 + 1 + 1
= 5 square units
The number of triangles is 5 and the sum of areas of the triangle is 5 square units.

Formula: 1 Mark
Value of n: 1 Mark
Numerical value: 1 Mark
Steps: 1 Mark
Given that,
${\left(a^m\right)}^{2n}=a^{2m}$
$a^{2mn}=a^{2m}$         $[\because {\left(a^m\right)}^n=a^{mn}]$
Since bases are same, their exponents also should be equal.
$\Rightarrow 2mn=2m$
$\Rightarrow n=\frac{2m}{2m}$
$\Rightarrow n=1$
As per question, 
a=2 and m=4
On substituting the values, we get:
 $a^{2m}$=$2^{2\times 4}$
                =$2^8$ 
                =$2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$
                =$256$
Hence, the required value is 256.

Steps: 2 Marks
Minimum Value: 1 Mark
Present Value: 1 Mark
Given that,
A's property = Rs 20
$\Rightarrow$ cube of A's property $={20}^3$
B's property = Rs 400
$\Rightarrow$ Square of B's property $={400}^2$
The minimum value of the man's property should be
$=Rs{20}^3\times {400}^2$
$=Rs128\times {10}^7$
$=Rs1.28\times {10}^9$
Given that
The actual value of the property is Rs2,00,00,00,000.
The value of the property depreciated by $30\%$.
The present value of the property$=2000000000-2000000000\times \frac{30}{100}$
$=2000000000-600000000$
$=1400000000$
Hence, the present value of the property is $Rs1400000000$

Each option: 2 Marks
a) Given that,
${\left(4^2\right)}^a={\left(7^a\right)}^2$  
$4^{2a}=7^{2a}$    $[{a^m}^n=a^{mn}]$
Given, $4^{2a}=7^{2a}$
Since $4\ne 7$ the only case where the above equation is true is when both the exponents are zero.
$\Rightarrow a=0$
$\Rightarrow 4^{2a}=7^{2a}=1$  $[\because 4^0=1,7^0=1]$
b) The given equation is
 $(2^a{)}^5=2^4\times 4^3$
$\Rightarrow 2^{a\times 5}=2^4\times (2^2{)}^3$
 
$\because (a^m{)}^n=a^{m\times n}$
$\Rightarrow 2^{a\times 5}=2^4\times \left(2^{2\times 3}\right)$
$\Rightarrow 2^{a\times 5}=2^4\times \left(2^6\right)$
$\Rightarrow 2^{a\times 5}=2^{4+6}=2^{10}$
$\because a^m\times a^n=a^{m+n}$
Since their bases are same and they are equal, threfore their powers must be same,
So, $5\times a=10$
$\Rightarrow a=\frac{10}{5}$
$\Rightarrow a=2$
So, the value of a = 2.

Each question: 2 Marks
i) It's easy to subtract powers when we convert them into normal form.
$0.005\times {10}^2=0.5$    (move the decimal point 2 places to the right)
$5\times {10}^{-1}=0.5$         (move the decimal point 1 place to the left)
Now,
$0.005\times {10}^2-5\times {10}^{-1}$
$=0.5-0.5$
$=0$
ii)
  $\frac{(2^5{)}^2\times 7^3}{8^3\times 7}$
= $\frac{(2^5{)}^2\times 7^3}{8^3\times 7}$
= $\frac{\left(2^{5\times 2}\right)\times 7^3}{(2^3{)}^3\times 7}$
= $\frac{2^{10}\times 7^3}{2^9\times 7}$
= $2^{10-9}\times 7^{3-1}=2\times 7^2$
= $2\times 49=98$

Steps: 2 Marks
Application: 1 Mark
Answer: 1 Mark
$6^{-5}\times 5^{-2}\times 6^3\times 5^m\times 6^n=1$
$6^{-5}\times 6^3\times 6^n\times 5^{-2}\times 5^m=1$
$6^{-5+3+n}\times 5^{-2+m}=1$
$6^{-2+n}\times 5^{-2+m}=1$
The bases of both the numbers are not equal.
So, the value will be equal to 1, when the exponents of these bases will be equal to zero.
Any non - zero number raised to the power zero is 1.
So, the exponents of both 6 and 5 are zero.
Therefore,
$-2+m=0\Rightarrow m=2$
$-2+n=0\Rightarrow n=2$
$\therefore m=n$

$81=3\times 3\times 3\times 3=3^4$

$625=5\times 5\times 5\times 5=5^4$

$\therefore \frac{81}{625}=\frac{3^4}{5^4}={\left(\frac{3}{5}\right)}^4$

${(-1)}^{evennumber}=+1$ 

Hence, ${\left(\frac{-3}{5}\right)}^4$  is also correct.

Putting $a=2$ and $b=3$,

we get $2^3=2\times 2\times 2=8$.

Here the  two terms  $5^3$ and $2^3$ have different bases, but same exponents.

$5^3\times 2^3=(5\times 5\times 5)\times (2\times 2\times 2)$
$=(5\times 2)\times (5\times 2)\times (5\times 2)$
$=(5\times 2{)}^3$
$={10}^3=1000$
or 
$5^3\times 2^3=(5\times 2{)}^3={10}^3=1000$