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24. Prove that the following are irrational:

(ii) $7 square root of 5$

(ii) $7 square root of 5$

Let us assume to the contrary, that $7 square root of 5$ is rational.
So we can find coprime integers a and b ( $space space not equal to$ 0 ) such that

$italic space italic space italic space italic space italic space italic space italic 7 square root of italic 5 italic equals a over b italic rightwards double arrow italic space italic space italic space italic space square root of italic 5 italic equals fraction numerator a over denominator italic 7 b end fraction$

Since, a and b are integers, $fraction numerator straight a over denominator 7 straight b end fraction$ is rational. and so, $square root of 5$ is rational

But this contradicts the fact that $square root of 5$ is irrational.

Therefore $7 square root of 5$ is irrational.

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25. Prove that the following are irrational:

(iii)

6 plus square root of 2

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26.

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

(i) $13 over 3125$

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27. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

(ii)

17 over 8

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28. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

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29. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion: