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Evaluate four-ninths to the power of three over two.
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First, we will need to recall the exponent power rule.
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This tells us that 𝑎 to the power of 𝑚 times 𝑛 is equal to 𝑎 to the power of 𝑚 to the power of 𝑛.
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And now since we can write three over two as one-half times three, we can use the exponent power rule in order to write four-ninths to the power of three over two is equal to four-ninths to the power of a half all to the power of three.
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Next, we will be using the exponent product rule.
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And this tells us that 𝑎 timesed by 𝑏 all to the power of 𝑚 is equal to 𝑎 to the 𝑚 timesed by 𝑏 to the 𝑚.
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And now we know that four-ninths can be written as four timesed by one over nine.
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And so we can write four-ninths to the power of a half as four to the power of a half timesed by one over nine to the power of a half.
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And this is also equal to four to the power of a half over nine to the power of a half.
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And we get that four-ninths to the power of three over two is also equal to four to the power of a half over nine to the power of a half all cubed.
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Now we can use the fact that 𝑥 to the power of a half is simply the square root of 𝑥.
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And so four to the power of a half is going to be equal to the square root of four.
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And nine to the power of a half is gonna be equal to the square root of nine.
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And we get that this is equal to the square root of four over the square root of nine all cubed.
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And now we are able to take the square root of four and nine, leaving us with two over three all cubed.
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Now it would not be wrong here to con- also consider the negative square root.
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However, we will only consider the positive one.
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Next, we can again use the exponent power rule to get that this is equal to two cubed over three cubed.
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And then since two cubed is eight and three cubed is 27, this gives us a final answer of eight over 27.
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Evaluate three over two to the power of minus three.
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First, we will use the exponent power rule, which, if we remember, tells us that 𝑎 to the power of 𝑚𝑛 is equal to 𝑎 to the power of 𝑚 times 𝑎 to the power of 𝑛.
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And since minus three is equal to minus one times three, we can write this as three over two to the power of minus one all cubed.
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Next, we will use the fact that 𝑥 to the power of minus one is equal to one over 𝑥.
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And so, therefore, three over two to the power of minus one is equal to two over three.
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And therefore, three over two to the power of minus three is equal to two over three cubed.
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Next, we will use the exponent product rule, which, if we remember, tells us that 𝑎 times 𝑏 to the power of 𝑚 is equal to 𝑎 to the power of 𝑚 timesed by 𝑏 to the power of 𝑚.
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Therefore, we get that two over three cubed is equal to two cubed over three cubed.
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Then since two cubed is equal to eight and three cubed is equal to 27, we get a final answer here of eight over 27.