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Evaluate 64 over 27 to the power of negative two-thirds.
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Well, the first thing we can do is split it up so that we deal with the numerator and denominator separately.
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And that’s because if we have a fraction that’s raised to the power of negative two over three, this is the same as 64 to the power of negative two over three or two-thirds over 27 to the power of negative two-thirds.
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So if we take a look at 64 to the power of negative two-thirds first, then to work this out what we’re gonna use is some of the things that we know about exponents.
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First of all, we know that 𝑥 to the power of negative 𝑎 is equal to one over 𝑥 to the power of 𝑎.
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We also know that 𝑥 to the power of one over 𝑎 is equal to the 𝑎th roots of 𝑥.
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And then one that’s gonna be very useful for this question is the fact that if we have 𝑥 to the power of 𝑎 over 𝑏, this is gonna be equal to 𝑥 to the power of one over 𝑏 all to the power of 𝑎 or 𝑥 to the power of 𝑎 to the power of one over 𝑏.
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And which version you use really depends on the number that you’ve got as the base.
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So, if we look at 64 to the power of negative two-thirds, then the way that we’re gonna split up is 64 to the power of a third to the power of negative two.
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And the reason we do that is because if we look at what we’ve got in the rules that we looked up, we can see that 64 to the power of a third is gonna be the cube root of 64.
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So, therefore, we’re gonna do this first because we know the cube root of 64 cause that’s just four.
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So, therefore, what we’re gonna be left with is four to the power of negative two, which is gonna give us one over four squared, which is gonna be one over 16.
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So, now, what we’re gonna look at is the denominator.
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And here we’ve got 27 to the power of negative two-thirds.
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So, once again, what we’re gonna do is put 27 to the power of a third all to the power of negative two.
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And that’s because we know the cube root of 27.
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And that’s three because three multiplied by three is nine, nine multiplied by three is 27.
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So, therefore, that’s gonna give us three to the power of negative two, which is gonna be equal to one over nine or one-ninth.
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And that’s because three to the power of negative two is one over three squared, and three squared is nine.
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So, now, what we have is one sixteenth over one-ninth, which we can rewrite as one sixteenth divided by one-ninth.
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So, now, what we want to do is divide by a fraction.
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So, therefore, what we have is a memory aid to help us remember how to do this.
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And that is KCF: Keep it, change it, flip it.
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So, if you keep here, we’re gonna keep the first fraction the same.
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Then, we’re gonna change the sign from a divide to multiply.
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And then, we’re gonna flip the second fraction.
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So that means we’re gonna get the reciprocal of it.
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So, instead of one over nine, we’ve got nine over one.
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So therefore, if we multiply one over 16 by nine over one, we’re gonna get nine over 16.
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So, therefore, we can say that if you evaluate 64 over 27 all to the power of negative two-thirds, you’re gonna get nine over 16 or nine sixteenths.