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Simplify π πΆ zero plus 17 multiplied by π πΆ one plus 17 squared multiplied by π πΆ two and so on plus 17 to the power of π multiplied by ππΆπ and so on up to 17 to the power of π multiplied by ππΆπ.
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In order to simplify this expression, we begin by recalling the binomial expansion of π plus π to the πth power.
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The first three terms of this expansion are π πΆ zero multiplied by π to the power of π plus π πΆ one multiplied by π to the power of π minus one multiplied by π plus π πΆ two multiplied by π to the power of π minus two multiplied by π squared.
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And the final term is ππΆπ multiplied by π to the πth power.
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We notice that much of this is the same as the expression in this question.
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Instead of π to the πth power, our last term contains 17 to the πth power.
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The second term contains 17 instead of π and the third term, 17 squared instead of π squared.
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This means that the value of π is 17.
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We notice that there is no π part to any of our terms.
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Since one raised to any power is equal to one, we can assume that π is equal to one as this is the only value for which this holds.
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The expression in the question is therefore equal to one plus 17 all raised to the πth power.
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And as one plus 17 equals 18, this is equal to 18 to the πth power.
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The expression π πΆ zero plus 17 multiplied by π πΆ one plus 17 squared multiplied by π πΆ two and so on up to 17 to the πth power multiplied by ππΆπ is equal to 18 to the πth power.