How do you simplify ${7}^{2}\times {7}^{6}$?

If you recall the way exponents are defined, you know that this means:

$\left(7\times 7\right)\times \left(7\times 7\times 7\times 7\times 7\times 7\right)$

If we remove the parentheses, we have the product of eight $7s$, which can be written more simply as:

${7}^{8}$

This suggests a shortcut: all we need to do is add the exponents!

${7}^{2}\times {7}^{6}={7}^{\left(\mathrm{2\; +\; 6}\right)}={7}^{8}$

In general, for all real numbers $a$, $b$, and $c$,

${a}^{b}\times {a}^{c}={a}^{\left(b+c\right)}$

To multiply two powers having the same base, add the exponents.