14 1 7 × 14 1 7 can be represented as ___________. Options: A . &nbsp 14 1 7 + 1 7 B . &nbsp ( 14 2 ) 1 7 C . &nbsp ( 14 × 14 ) 1 7 D . &nbsp All of the above

Answer: Option D : D For any positive real number ' a ' and integers ' m ' and ' n ', we define 1. a m × a n = a m + n 2. ( a m ) n = a m n 3. ( a b ) m = a m b m and if a = b , the expression becomes ( a × a ) m = ( a 2 ) m = a 2 m . So, using first rule, we have 14 1 7 + 1 7 = 14 2 7 Now, using second rule, ( 14 2 ) 1 7 = 14 2 7 Finally, using third rule, ( 14 × 14 ) 1 7 = 14 2 7 Hence, we see that all the three results are same.

1417×1417 can be represented as __________

Question

$14^{\frac{1}{7}} \times 14^{\frac{1}{7}}$ can be represented as ___________.

Options:

A .

14^{\frac{1}{7} + \frac{1}{7}}

B .
(

14^{2})^{\frac{1}{7}}

C . (

14 \times 14)^{\frac{1}{7}}

D . All of the above

Answer: Option D
:
D

For any positive real number ' $a$ ' and integers ' $m$ ' and ' $n$ ', we define
1. $a^{m} \times a^{n} = a^{m + n}$
2. ${(a^{m})}^{n} = a^{m n}$
3. $(a b)^{m} = a^{m} b^{m}$ and if $a = b$ , the expression becomes $(a \times a)^{m} = (a^{2})^{m} = a^{2 m}$ .
So, using first rule, we have
$14^{\frac{1}{7} + \frac{1}{7}} = 14^{\frac{2}{7}}$
Now, using second rule,
$(14^{2})^{\frac{1}{7}}$ = $14^{\frac{2}{7}}$
Finally, using third rule,
( $14 \times 14)^{\frac{1}{7}} = 14^{\frac{2}{7}}$
Hence, we see that all the three results are same.