Simplify and write in exponential form:
\[\begin{align}
  & (1){{2}^{5}}\times {{2}^{3}} \\
 & (2){{p}^{3}}\times {{p}^{2}} \\
 & (3){{4}^{3}}\times {{4}^{2}} \\
 & (4){{a}^{3}}\times {{a}^{2}}\times {{a}^{7}} \\
 & (5){{5}^{3}}\times {{5}^{7}}\times {{5}^{12}} \\
 & (6){{(-4)}^{100}}\times {{(-4)}^{2}} \\
\end{align}\]

Hint: Exponential means a mathematical expression that has one or more exponents. It is a way of representing repeated multiplications of the same number. By writing the number as a base with the number of repeats written as a small number to its upper right.
Example- 32 can be written as \[{{2}^{5}}\]

Formula Used:
 \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
\[{{a}^{m}}\times {{a}^{n}}\times {{a}^{p}}={{a}^{m+n}}^{+p}\]

Complete step-by-step answer:
\[(1){{2}^{5}}\times {{2}^{3}}\]
using the formula \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
here ,
a=2, m=5 and n=3
\[\Rightarrow \]\[{{2}^{5}}\times {{2}^{3}}\] can be written as \[{{2}^{5+3}}\]
Hence, \[{{2}^{5}}\times {{2}^{3}}\]=\[{{2}^{5+3}}\]=\[{{2}^{8}}\]

\[(2){{p}^{3}}\times {{p}^{2}}\]
using the formula \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
here ,
a=p, m=3 and n=2
\[\Rightarrow \]\[{{p}^{3}}\times {{p}^{2}}\] can be written as \[{{p}^{3+2}}\]
Hence, \[{{p}^{3}}\times {{p}^{2}}\]=\[{{p}^{3+2}}\]=\[{{p}^{5}}\]

\[(3){{4}^{3}}\times {{4}^{2}}\]
using the formula \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
here ,
a=4, m=3 and n=2
\[\Rightarrow \]\[{{4}^{3}}\times {{4}^{2}}\] can be written as \[{{4}^{3+2}}\]
Hence, \[{{4}^{3}}\times {{4}^{2}}\]=\[{{4}^{3+2}}\]=\[{{4}^{5}}\]
(4)\[{{a}^{3}}\times {{a}^{2}}\times {{a}^{7}}\]
Using the formula \[{{a}^{m}}\times {{a}^{n}}\times {{a}^{p}}={{a}^{m+n}}^{+p}\]
here ,
a=a, m=3 , n=2 and p=7
\[\Rightarrow \]\[{{a}^{3}}\times {{a}^{2}}\times {{a}^{7}}\] can be written as\[{{a}^{3+2+7}}\]
Hence, \[{{a}^{3}}\times {{a}^{2}}\times {{a}^{7}}\]=\[{{a}^{3+2+7}}\]=\[{{a}^{12}}\]

(5)\[{{5}^{3}}\times {{5}^{7}}\times {{5}^{12}}\]
Using the formula \[{{a}^{m}}\times {{a}^{n}}\times {{a}^{p}}={{a}^{m+n}}^{+p}\]
here ,
a=5, m=3 , n=7 and p=12
\[\Rightarrow \]\[{{5}^{3}}\times {{5}^{7}}\times {{5}^{12}}\] can be written as\[{{5}^{3+7+12}}\]
Hence, \[{{5}^{3}}\times {{5}^{7}}\times {{5}^{12}}\]=\[{{5}^{3+7+12}}\]=\[{{5}^{22}}\]

(6)\[{{(-4)}^{100}}\times {{(-4)}^{2}}\]
using the formula \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
here ,
a=(-4), m=100 and n=2
\[\Rightarrow \]\[{{(-4)}^{100}}\times {{(-4)}^{2}}\] can be written as \[{{(-4)}^{100+2}}\]
Hence, \[{{(-4)}^{100}}\times {{(-4)}^{2}}\]=\[{{(-4)}^{100+2}}\]=\[{{(-4)}^{102}}\]=\[{{(-1)}^{102}}\times {{4}^{102}}\] { using formula \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]}
We know, \[{{(-1)}^{102}}=1\] {as power of (-1) is even }
So, \[{{(-4)}^{100}}\times {{(-4)}^{2}}\]=\[{{4}^{102}}\]
Additional Information: Let us take the example of the number 1000.
We can write 1000 as 10x10x10, but instead of writing 10 three times we can write the number 1000 in an alternative way too.
1000 = 10 x 10 x 10
1000 = (10)3………..(exponential form)
Let us take another example to further clarify the concept of expressing a number in exponential form.
Let us take the example of a number 576895.
Now we can express this number in the following ways
576895 = 5×100000 + 7×10000 + 6×1000 + 8×100 + 9×10 + 5×1
576895= 5 x 105 + 7×104 + 6×103 + 8×102 + 9×101 + 5x 100
The last term which includes 100 can simply be written as 1 because any number when expressed raising it to the power of 0 it equals 1.
Therefore we can clearly say that exponents simply refer to how many times a particular number appears in a numerical term.
Some additional formulas that we need to know,
\[\begin{align}
  & {{a}^{m}}/{{a}^{n}}={{a}^{m-n}} \\
 & {{a}^{m}}\times {{b}^{m}}={{(a\times b)}^{m}} \\
\end{align}\]

Note: The knowledge about the exponential is important for students to answer such types of questions. The additional information of factors of different numbers is also required to answer such questions