What is the value of nine to the 𝑥 minus one power times three to the four 𝑥 plus two power times one-third to the four 𝑥 power if three to the 𝑥 power equals eight?
We’re told that three to the 𝑥 power equals eight. And this is the expression we want to solve. Now one strategy that we sometimes use to solve for exponents would be to rewrite eight as a base three. It would be to say eight is equal to three to the what power. Unfortunately, we know that three to the first power equals three and three squared equals nine. And that means the exponent here will not be an integer. And so we’ll need a different strategy.
If we go and look at the expression, we can see that it seems like all three of these bases are some kind of factor of three. We could try to then write all three of these values with a base three. For example, nine equals three squared. And so we could write nine to the 𝑥 minus one power as three squared to the 𝑥 minus one power. We’ll just bring down this three to the four 𝑥 plus two power.
What about one-third? How do we write one-third with a base three? We can think about one-third as being one over three to the first power. And we know that one over 𝑎 to the 𝑥 power equals 𝑎 to the negative 𝑥 power, which means one-third can be written as three to the negative one power. And one-third to the four 𝑥 power is then three to the negative one power to the four 𝑥 power.
We should think about what it means to take a power of a power. 𝑎 to the 𝑥 power to the 𝑦 power is equal to 𝑎 to the 𝑥 times 𝑦 power. And that means we could multiply two times 𝑥 minus one to get three to the two 𝑥 minus two power times three to the four 𝑥 plus two power. And then if we multiply negative one by four 𝑥, we get three to the negative four 𝑥 power.
We need another exponent rule at this point. 𝑎 to the 𝑥 power times 𝑎 to the 𝑦 power is equal to 𝑎 to the 𝑥 plus 𝑦 power. Because all three of these values have a base three and are being multiplied together, we can do that by taking the base three and then adding all the powers. Two 𝑥 minus two plus four 𝑥 plus two minus four 𝑥. We have minus two plus two, which equals zero. And we have plus four 𝑥 minus four 𝑥, which equals zero.
We’ve simplified our expression to be three to the two 𝑥 power. But we don’t know what that is. We only know what three to the 𝑥 power is. And so we want to use this exponent rule of a power to a power to see if we could rewrite three to the two 𝑥 power in a format that we can use. We could rewrite three to the two 𝑥 power as three to the 𝑥 power squared.
And at that point, we know that three to the 𝑥 power equals eight. And our expression is three to the 𝑥 power squared. Eight squared equals 64. Therefore, the value of our given expression is 64.