Question Video: Simplifying Expansions Using the Binomial Theorem | Nagwa Question Video: Simplifying Expansions Using the Binomial Theorem | Nagwa

# Question Video: Simplifying Expansions Using the Binomial Theorem Mathematics • Third Year of Secondary School

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Simplify ππΆ0 + 17 Γ ππΆ1 + 17Β² Γ ππΆ2+ ... + 17^(π) Γ ππΆπ + ... + 17 ^(π) Γ ππΆπ.

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### Video Transcript

Simplify π πΆ zero plus 17 multiplied by π πΆ one plus 17 squared multiplied by π πΆ two and so on plus 17 to the power of π multiplied by ππΆπ and so on up to 17 to the power of π multiplied by ππΆπ.

In order to simplify this expression, we begin by recalling the binomial expansion of π plus π to the πth power. The first three terms of this expansion are π πΆ zero multiplied by π to the power of π plus π πΆ one multiplied by π to the power of π minus one multiplied by π plus π πΆ two multiplied by π to the power of π minus two multiplied by π squared. And the final term is ππΆπ multiplied by π to the πth power. We notice that much of this is the same as the expression in this question. Instead of π to the πth power, our last term contains 17 to the πth power. The second term contains 17 instead of π and the third term, 17 squared instead of π squared. This means that the value of π is 17.

We notice that there is no π part to any of our terms. 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. The expression in the question is therefore equal to one plus 17 all raised to the πth power. And as one plus 17 equals 18, this is equal to 18 to the πth power.

The expression π πΆ zero plus 17 multiplied by π πΆ one plus 17 squared multiplied by π πΆ two and so on up to 17 to the πth power multiplied by ππΆπ is equal to 18 to the πth power.

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