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What is the value of nine to the π₯ minus one power times three to the four π₯ plus two power times one-third to the four π₯ power if three to the π₯ power equals eight?
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Weβre told that three to the π₯ power equals eight.
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And this is the expression we want to solve.
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Now one strategy that we sometimes use to solve for exponents would be to rewrite eight as a base three.
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It would be to say eight is equal to three to the what power.
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Unfortunately, we know that three to the first power equals three and three squared equals nine.
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And that means the exponent here will not be an integer.
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And so weβll need a different strategy.
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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.
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We could try to then write all three of these values with a base three.
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For example, nine equals three squared.
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And so we could write nine to the π₯ minus one power as three squared to the π₯ minus one power.
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Weβll just bring down this three to the four π₯ plus two power.
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What about one-third?
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How do we write one-third with a base three?
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We can think about one-third as being one over three to the first power.
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And we know that one over π to the π₯ power equals π to the negative π₯ power, which means one-third can be written as three to the negative one power.
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And one-third to the four π₯ power is then three to the negative one power to the four π₯ power.
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We should think about what it means to take a power of a power.
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π to the π₯ power to the π¦ power is equal to π to the π₯ times π¦ power.
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And that means we could multiply two times π₯ minus one to get three to the two π₯ minus two power times three to the four π₯ plus two power.
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And then if we multiply negative one by four π₯, we get three to the negative four π₯ power.
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We need another exponent rule at this point.
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π to the π₯ power times π to the π¦ power is equal to π to the π₯ plus π¦ power.
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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.
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Two π₯ minus two plus four π₯ plus two minus four π₯.
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We have minus two plus two, which equals zero.
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And we have plus four π₯ minus four π₯, which equals zero.
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Weβve simplified our expression to be three to the two π₯ power.
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But we donβt know what that is.
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We only know what three to the π₯ power is.
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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 π₯ power in a format that we can use.
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We could rewrite three to the two π₯ power as three to the π₯ power squared.
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And at that point, we know that three to the π₯ power equals eight.
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And our expression is three to the π₯ power squared.
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Eight squared equals 64.
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Therefore, the value of our given expression is 64.