Video Transcript
If nine to the power of 𝑥 equals six, what is nine to the power of 𝑥 minus one?
To answer this question, we’re going to take nine to the power of 𝑥 minus one and see if we can write it in the form involving nine to the power of 𝑥 and some other terms. We can recall the exponent rule 𝑥 to the power of 𝑎 plus 𝑏 is equal to 𝑥 to the power of 𝑎 times 𝑥 to the power of 𝑏. So, taking the term nine to the power of 𝑥 minus one, we can rewrite this as nine to the power of 𝑥 times nine to the power of negative one. We must be very careful to include the negative sign alongside one.
Noticing now that we have nine to the power of 𝑥, which we’re told is equal to six, we can substitute this value in. Which gives us nine to the power of 𝑥 minus one is equal to six times nine to the power of negative one. Taking a closer look at the value nine to the power of negative one, we can use the fact that any number to the power of negative one simply means take the reciprocal of it. In this case, 𝑥 to the power of negative one is equivalent to one over 𝑥.
So, our nine to the power of negative one is equivalent to the fraction one-ninth. We can then multiply this by six to give us six-ninths. Simplifying this will give us our final answer that nine to the power of 𝑥 minus one is equal to two-thirds.
