- Thread starter megan0430
- Start date

Hello, Megan!

If I interpreted your notation correctly, I have the solution . . .

\(\displaystyle \text{Find }x:\;\;\L a^{^{x^2}}\cdot a^{^{6x}}\:=\:a^{-8}\;\)\(\displaystyle \text{ for }a\,\neq\,0\)

Multiply both sides by \(\displaystyle a^8:\)\(\displaystyle \L\;\;\left(a^{^{x^2}}\right)\left(a^{^{6x}}\right)\left(a^{^8})\;=\;1\)

\(\displaystyle \;\;\)This becomes: \(\displaystyle \L\:a^{^{(x^{^2}+6x+8)}}\;=\;1\;=\;a^{^0}\)

And we have the quadratic: \(\displaystyle \L\:x^2\,+\,6x\,+\,8 \:=\:0\)

\(\displaystyle \;\;\)which factors: \(\displaystyle \L\x\,+\,2)(x\,+\,4)\;=\;0\)

\(\displaystyle \;\;\) and has roots: \(\displaystyle \L\:x\:=\:-2,\;-4\)