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Complete the following: The fifth root of 243 is equal to the cube root of what.
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The first thing to realize here is that we have a problem involving đť‘›th roots.
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On the left-hand side, we have the fifth root of 243.
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Notice that this five is written smaller within the root sign.
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Itâ€™s different to five multiplied by the square root of 243.
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This is not what we have here.
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We have a fifth root.
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So letâ€™s approach this problem by seeing if we can work out what the left-hand side would be equal to.
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In order to find the fifth root of 243, we could say that thereâ€™s some value â€” letâ€™s call it đť‘Ą â€” and when we take that to the fifth power, that would give us 243.
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Letâ€™s consider what value đť‘Ą could be.
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We know that it canâ€™t be one since one to the power five would give us one.
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And đť‘Ą couldnâ€™t be two since any exponent of two will give us an even value.
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So letâ€™s see if đť‘Ą could be three.
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Does three to the power of five give us 243?
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Well, three times three would give us nine.
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And then when we multiply that by the third three, that would give us 27.
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Then, multiplied by the fourth three, that would give us 81.
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And finally, 81 multiplied by the final three would indeed give us 243.
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Now, we can say that since three to the power of five gives us 243, then the fifth root of 243 is three.
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That means that weâ€™ve simplified the left-hand side of this equation.
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So letâ€™s take a look at the right-hand side.
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The blank that weâ€™re missing is some value that when we take the cube root of it, it gives us three.
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In order to solve this equation, weâ€™d need to perform the inverse operation.
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In this case, the inverse operation to taking the cube root is cubing or finding the third power.
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Three to the power of three is three times three times three, which gives us 27.
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Therefore, the missing value must be 27.
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In order to check that this answer of 27 would be correct, we can remember that the fifth root of 243 is three.
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So thatâ€™s the left-hand side.
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And on the right-hand side, the cube root of 27 would also give us three.
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Since the two sides of the equation are equal, then 27 must be the correct answer.