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Complete the following: The sixth root of 64 equals the cube root of blank.
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In this question, on the right-hand side, we have an unknown value which has the third or the cube root taken off it.
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The cube root of this unknown value is equal to the sixth root of 64.
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So, perhaps the best place to start is to actually work out the sixth root of 64.
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If we say that the sixth root of 64 would give us this value, which we can call 𝑥, it’s equivalent to writing 𝑥 six times and multiplying.
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And that would give us a value of 64.
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The question is, what is 𝑥?
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If we were to try some different values of 𝑥 and we started with one, well, one written six times and multiplied would still give us a value of one, which isn’t the value of 64 that we were looking for.
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So, how about a value of 𝑥 equals two?
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There’s a number of different ways to work this out, but we need to be careful that we’re not multiplying six by two.
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If we began by multiplying the first two times two, that would give us an answer of four.
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And we could then multiplied it by the next two, which would give us an answer of eight.
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We might then notice that we have another two times two times two, which would also give us eight, and multiplying eight times eight would give us 64.
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That means that we have worked out that the sixth root of 64 is two.
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And so, the left-hand side of our equation should be equal to two, and so should the right-hand side.
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We should be careful when we’re answering questions like this, as if we just wrote in the value of two on the missing line, it would be incorrect.
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Since this expression on the right-hand side is equal to two, we need to think what value taken to the third root would give us an answer of two.
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The inverse of finding the third root is to find the third power, so we need to calculate two to the third power.
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And it’s eight since two times two times two is eight.
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We can, therefore, give the missing answer as eight since the sixth root of 64 is equal to two and the cube root of eight is also equal to two.