Question

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

• 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 1

The correct option is D

All of the above

For any positive real number '$a$' and integers '$m$' and '$n$', we define
1. $a^m\times a^n=a^{m+n}$
2. ${\left(a^m\right)}^n=a^{mn}$
3. $(ab{)}^m=a^m{b}^m$ and if $a=b$, the expression becomes $(a\times a{)}^m=(a^2{)}^m=a^{2m}$.

​​​​​​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.